x = -1 and y = 3 in equations (i) and (ii):x + y = 2-1 + 3 = 2 (satisfied)3x + y = 0-3 + 3 = 0 (satisfied)
a) Solving the system using substitution:
We know that: x+y=2 (i)3x+y=0 (ii)We will solve equation (i) for y:y=2-x
Now, substitute this value of y in equation (ii):3x + (2-x) = 03x+2-x=0 2x = -2 x = -1
Substitute the value of x in equation (i):x + y = 2-1 + y = 2y = 3b)
Solving the system using elimination (linear combination) :
We know that: x+y=2 (i)3x+y=0 (ii)
We will subtract equation (i) from equation (ii):3x + y - (x + y) = 0 2x = 0 x = 0
Substitute the value of x in equation (i):0 + y = 2y = 2c)
Solving the system by graphing:We know that: x+y=2 (i)3x+y=0 (ii)
Let us plot the graph for both the equations on the same plane:
graph{x+2=-y [-10, 10, -5, 5]}
graph{y=-3x [-10, 10, -5, 5]}
From the graph, we can see that the intersection point is (-1, 3)d)
We calculated the value of x and y in parts a, b, and c and the solutions are as follows:
Substitution: x = -1, y = 3
Elimination: x = 0, y = 2
Graphing: x = -1, y = 3
We can see that the value of x is different in parts a and b but the value of y is the same.
The value of x is the same in parts a and c but the value of y is different.
However, the value of x and y in part c is the same as in part a.
Therefore, we can say that the solutions of parts a, b, and c are not the same.
However, we can check if these solutions satisfy the original equations or not. We will substitute these values in the original equations and check:
Substituting x = -1 and y = 3 in equations (i) and (ii):x + y = 2-1 + 3 = 2 (satisfied)3x + y = 0-3 + 3 = 0 (satisfied)
Therefore, the values we obtained for x and y are the correct solutions.
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F(x)=1−e^−x^2 ;−[infinity]≤x<0.
(ii) F(x)=1−e^−x^2 ;0≤x<[infinity].
(b) For the F(x) in one of parts (i) and (ii) of (a) that represents a valid cumulative distribution function, (i) Derive the probability density function of X. (ii) Derive the median of X. (iii) Derive the 95-th percentile of X. (iv) Derive the expectation of X. (v) Derive the moment generating function of X.
(ii) The median of X is ±√(-ln(0.5)).
(iii) The 95th percentile of X is ±√(-ln(0.05)).
(iv) E(X) = -x * [tex]e^(-x^2)[/tex]+ √π/2
(v) MGF is not possible in this case.
Let's work through the parts of the problem one by one:
(a) First, let's determine which part of the function represents a valid cumulative distribution function (CDF).
(i) F(x) = 1 - [tex]e^(-x^2)[/tex]; -∞ ≤ x < 0
(ii) F(x) = 1 - [tex]e^(-x^2)[/tex]; 0 ≤ x < ∞
To be a valid CDF, the function F(x) must satisfy the following conditions:
1. F(x) is non-decreasing.
2. F(x) is right-continuous.
3. lim(x→-∞) F(x) = 0 and lim(x→∞) F(x) = 1.
Both parts (i) and (ii) satisfy these conditions, so both can represent valid CDFs. We'll continue with part (ii) for the subsequent calculations.
(i) Derive the probability density function of X:
To derive the probability density function (PDF), we need to take the derivative of the cumulative distribution function (CDF) with respect to x.
f(x) = d/dx [F(x)]
f(x) = d/dx [1 - [tex]e^(-x^2)[/tex]]
To find the derivative, we'll use the chain rule:
f(x) = -2x * [tex]e^(-x^2)[/tex]
(ii) Derive the median of X:
The median of a distribution is the value of x for which F(x) = 0.5.
Setting F(x) = 0.5 and solving for x:
0.5 = 1 - [tex]e^(-x^2)[/tex]
[tex]e^(-x^2)[/tex] = 0.5
-x² = ln(0.5)
x² = -ln(0.5)
x = ±√(-ln(0.5))
The median of X is ±√(-ln(0.5)).
(iii) Derive the 95th percentile of X:
The pth percentile of a distribution is the value [tex]x_p[/tex] such that F[tex](x_p)[/tex] = p.
Setting [tex]F(x_p)[/tex] = 0.95 and solving for [tex]x_p[/tex]:
0.95 = 1 - [tex]e^(-x_p^2)[/tex]
[tex]e^(-x_p^2)[/tex] = 0.05
[tex]-x_p^2[/tex] = ln(0.05)
[tex]x_p^2[/tex] = -ln(0.05)
[tex]x_p[/tex]= ±√(-ln(0.05))
The 95th percentile of X is ±√(-ln(0.05)).
(iv) Derive the expectation of X:
The expectation of X, denoted as E(X), is given by the integral of x times the PDF over its entire range.
E(X) = ∫x * f(x) dx, where the integral is taken from -∞ to ∞.
E(X) = ∫x * (-2x * [tex]e^(-x^2)[/tex]) dx
To solve this integral, we can use integration by parts:
Let u = x, dv = -2x * [tex]e^(-x^2)[/tex]dx
Then, du = dx, v = [tex]-e^(-x^2)[/tex]
Using the integration by parts formula:
∫u * dv = uv - ∫v * du
E(X) = [-x * [tex]e^(-x^2)[/tex]] - ∫[tex](-e^(-x^2))[/tex]dx
E(X) = -x * [tex]e^(-x^2)[/tex] + ∫[tex]e^(-x^2)[/tex] dx
To evaluate the integral ∫[tex]e^(-x^2)[/tex]dx, we can use the Gaussian integral, which does not have a simple
closed-form solution. The result of this integral is √π/2.
E(X) = -x * [tex]e^(-x^2)[/tex]+ √π/2
(v) Derive the moment generating function of X:
The moment generating function (MGF) of a random variable X is defined as the expected value of [tex]e^(tX)[/tex], where t is a parameter.
M(t) = E[tex](e^(tX)[/tex])
To find the MGF, we substitute the expression for X into the MGF formula:
M(t) = E[tex](e^(tX)[/tex])
M(t) = ∫[tex]e^(tx)[/tex] * f(x) dx
Using the PDF derived earlier:
M(t) = ∫[tex]e^(tx) * (-2x * e^(-x^2))[/tex] dx
We can simplify this expression by multiplying the terms:
M(t) = -2 ∫x * [tex]e^(tx - x^2)[/tex] dx
Finding the integral in closed form is challenging due to the combination of exponential and quadratic terms. However, we can still compute the MGF numerically or approximate it using techniques like Taylor series expansion.
Note: Due to the complexity of the integral involved, finding a closed-form solution for the expectation and MGF is not possible in this case.
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2. Show that \( \vec{\alpha}(t)=\left(t, 1+\frac{1}{t}, \frac{1}{t}-t\right), t>0 \) lies on a plane \( P \). \( P= \) ?
There are no values of \( t \) that satisfy all the equations simultaneously. This means that the vector function \( \vec{\alpha}(t) \) does not lie on any plane \( P \).
To show that the vector function \( \vec{\alpha}(t) = \left(t, 1+\frac{1}{t}, \frac{1}{t}-t\right) \) lies on a plane \( P \), we need to find the equation of the plane.
Let's express the vector \( \vec{\alpha}(t) \) as a linear combination of two linearly independent vectors that lie in the plane \( P \). We can choose two vectors \( \vec{v_1} \) and \( \vec{v_2} \) that are not collinear. The plane \( P \) can be represented by the equation:
\( \vec{r} = \vec{a} + s\vec{v_1} + t\vec{v_2} \),
where \( \vec{r} \) is a vector in the plane, \( \vec{a} \) is a position vector to a point on the plane, and \( s \) and \( t \) are scalars.
We can rewrite \( \vec{\alpha}(t) \) as:
\( \vec{\alpha}(t) = t\vec{i} + \left(1+\frac{1}{t}\right)\vec{j} + \left(\frac{1}{t}-t\right)\vec{k} \).
Comparing the coefficients of \( \vec{i} \), \( \vec{j} \), and \( \vec{k} \), we have:
\( t = s \) (coefficient of \( \vec{v_1} \))
\( 1+\frac{1}{t} = t \) (coefficient of \( \vec{v_2} \))
\( \frac{1}{t}-t = 0 \) (coefficient of \( \vec{k} \))
From the third equation, we get \( \frac{1}{t} = t \), which implies \( t = \pm 1 \). Since \( t > 0 \), we have \( t = 1 \).
Substituting \( t = 1 \) into the second equation, we have:
\( 1 + \frac{1}{1} = 1 \),
which simplifies to \( 2 = 1 \), which is not true.
Therefore, there are no values of \( t \) that satisfy all the equations simultaneously. This means that the vector function \( \vec{\alpha}(t) \) does not lie on any plane \( P \).
In conclusion, the vector \( \vec{\alpha}(t) = \left(t, 1+\frac{1}{t}, \frac{1}{t}-t\right) \) does not lie on a plane \( P \).
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A survey found that women's heights are normally distributed with mean 63.1 in. and standard deviation 2.1 in. The survey also found that men's heights are normally distributed with mean 67.1 in. and standard deviation 3.1 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 56in. and a maximum of 63 in. Complete parts (a) and (b) below. a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement park? The percentage of men who meet the height requirement is %. (Round to two decimal places as needed.)
a) The percentage of men meeting the height requirement is %Here, Mean height of men, μ1 = 67.1 in and Standard deviation, σ1 = 3.1 in Minimum height requirement, x1 = 56 in and Maximum height requirement, x2 = 63 in To find the percentage of men meeting the height requirement, we need to find the z-scores for minimum height requirement and maximum height requirement.
Using a standard normal distribution table, we can find that the area to the left of z1 is 0.0002 and area to the left of z2 is 0.0918.The percentage of men meeting the height requirement is (0.0918 - 0.0002) × 100 = 9.16%.The result suggests that the percentage of men meeting the height requirement is very small compared to the women who are employed as characters in the amusement park.
Since men have a higher mean and a higher standard deviation than women, it is not surprising that fewer men are able to meet the height requirements of the amusement park. b) The percentage of women meeting the height requirement is % Here, Mean height of women, μ2 = 63.1 in and Standard deviation, σ2 = 2.1 in Minimum height requirement.
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Find y as a function of x if y′′−9y′′+18y′=50e^2
The auxiliary equation of the given differential equation: [tex]y^2-9y+18 = 0[/tex]Solving for y:
Now, we find the complementary function:
[tex]Y_c = c_1 * e^2x + c_2 * e^3x[/tex]
We can observe that the right side of the equation is of the form of the non-homogeneous part which is 50e^(2x).
Therefore, we assume the particular solution of the form:
[tex]Y_p = Ae^(2x)[/tex]where A is a constant.
Taking the derivatives of[tex]y_p: y_p' = 2Ae^(2x)y_p'' = 4Ae^(2x)[/tex]
Putting the values of [tex]y_p, y_p', and y_p''[/tex]in the given differential equation,
We have:
[tex]4Ae^(2x) - 9(2Ae^(2x)) + 18(2Ae^(2x)) = 50e^(2x)[/tex]
Simplifying the equation:[tex]4A - 18A + 36A = 50A = 1.25[/tex]
The particular solution is:[tex]Y_p = 1.25e^(2x)[/tex]
Hence, the general solution to the given differential equation:[tex]y(x) = Y_c + Y_p[/tex]
Where Y_c is the complementary function and Y_p is the particular solution obtained.
Therefore,[tex]y(x) = c_1 * e^2x + c_2 * e^3x + 1.25e^(2x)[/tex]
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Suppose each person has 12 hours for the two tasks in a week, and suppose both Sheldon and Leonard each spend 6 hours on cooking and 6 hours on laundry. Consider an offer from Leonard to Sheldon: do 3 baskets of laundry for me each week, and I’ll cook you 2 meals. Can you find a production plan such that the offer benefits both Leonard and Sheldon? Hint: For a production plan, you need to specify how each person divides his 12 hours between the two tasks. The offer benefits Sheldon (or Leonard) when it results in no fewer meals and no fewer baskets for him. (10 points)
1.8. Consider the same setup in 1.7. Consider another offer from Leonard to Sheldon: do 1 basket of laundry for me each week, and I’ll cook you 3 meals. Can you find a production plan such that the offer benefits both Leonard and Sheldon? Explain your answer.
Both offers are mutually beneficial and help to save time for both parties.
Here, the production plans for Leonard and Sheldon for both offers can be as follows:
Offer 1: Leonard does 7 hours of cooking and 5 hours of laundry, while Sheldon does 5 hours of cooking and 7 hours of laundry. This way, Leonard gets to save 1 hour on laundry and Sheldon gets 1 extra meal.
Offer 2: Leonard does 9 hours of cooking and 3 hours of laundry, while Sheldon does 3 hours of cooking and 9 hours of laundry. This way, Leonard gets to save 3 hours on cooking and Sheldon gets 2 extra meals.
Leonard and Sheldon spend equal amounts of time on cooking and laundry, i.e., 6 hours on each. Leonard offers Sheldon to do his laundry in return for cooking him more meals.
In the first offer, Sheldon does 7 hours of laundry and 5 hours of cooking, while Leonard does 5 hours of laundry and 7 hours of cooking.
Thus, Sheldon gets 1 extra meal and Leonard saves 1 hour on laundry.
In the second offer, Sheldon does 9 hours of laundry and 3 hours of cooking, while Leonard does 3 hours of laundry and 9 hours of cooking.
Thus, Sheldon gets 2 extra meals and Leonard saves 3 hours on cooking. Therefore, both offers benefit each person in their own way.
Thus, we can conclude that in the first offer, Sheldon gets to eat one extra meal while Leonard gets to save an hour on laundry. In the second offer, Sheldon gets to eat two extra meals while Leonard saves three hours on cooking. Therefore, both offers are mutually beneficial and help to save time for both parties.
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A hiker walks with an average speed of 2.4 m/s. What distance in kilometers does the hiker travel in a time of 2.6 hours? The distance traveled by the hiker in a time of 2.6 hours is km.
The distance traveled by the hiker is 22.464 km = 22.464 × 1000 = 22464 m. Rounded off to two decimal places, the distance traveled by the hiker in 2.6 hours is 6.24 km.
The distance traveled by the hiker in a time of 2.6 hours is 6.24 km.
Given data: Average speed of the hiker = 2.4 m/s, Time taken by the hiker to travel = 2.6 hours
First, we need to convert the average speed from m/s to km/h.1 m/s = 3.6 km/h
Therefore, the average speed of the hiker = 2.4 m/s × 3.6 = 8.64 km/h
Now, we can use the formula distance = speed × time to find the distance traveled by the hiker.
Distance = Speed × Time
Distance = 8.64 km/h × 2.6 h = 22.464 km
However, the distance is required in kilometers, not in meters.
Therefore, we convert km to meters.1 km = 1000 m
Hence, the distance traveled by the hiker is 22.464 km = 22.464 × 1000 = 22464 m.
To convert the above value to kilometers, we need to divide it by 1000 (since 1 km = 1000 m).Distance in kilometers = Distance in meters ÷ 1000Distance in kilometers = 22464 m ÷ 1000 = 22.464 km
Rounded off to two decimal places, the distance traveled by the hiker in 2.6 hours is 6.24 km.
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Peter, Paul and Mary are having a threesome tug of war. Peter pulls to the left with a force of 15.7 pounds and Paul pulls to the right with a force of 17.8 pounds. In order for the game to be fair, and balanced, what team should Mary be on and what force must she pull with?
Mary should join Peter's team and pull to the left with a force of 2.1 pounds. This would create equilibrium, with a total force of 15.7 pounds on each side.
To determine the team on which Mary should be and the force she must pull with to create a fair and balanced tug of war game, we need to consider the forces exerted by Peter and Paul.
Let's analyze the situation:
Peter's force: 15.7 pounds (pulling to the left)
Paul's force: 17.8 pounds (pulling to the right)
To create balance, Mary's force should equalize the total force exerted on each side. In this case, Peter's force and Paul's force need to be equalized.
Since Peter is pulling with 15.7 pounds to the left and Paul is pulling with 17.8 pounds to the right, there is a difference of 2.1 pounds between the forces.
For the game to be fair and balanced, Mary should join Peter's team and pull to the left with a force of 2.1 pounds. This would create equilibrium, with a total force of 15.7 pounds on each side.
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There is a very useful equation for predicting diffusion times and distances for a semi-infinite solid. The equation comes from our simplification of the error function, and allows us to write:
x = (Dt)1/2
Where x is the diffusion depth, D is the diffusion coefficient, and t is the time of the diffusion process. Use this relationship to solve the problem below:
Figure: Engine timing gear from a Ducati superbike carburized to a depth of 0.8 mm
After a 10 hour treatment of a steel gear at 925 oC, the carbon diffusion depth is evaluated to be 0.4 mm. The gear requires a depth of 0.8mm for the intended application. Predict how long it will take for the same sample at the same processing temperature to reach the target carbon diffusion depth? (if you say 20 hours I may laugh at you when I see your homework…). Use a value of 2.00 x 10-4 m2/s for Do.
According to the given problem statement,[tex]D = 2.00 x 10-4 m²/st = 0.2 x 10-3 cm²/s,[/tex] the carbon diffusion depth is evaluated to be 0.4 mm and the gear requires a depth of 0.8 mm for the intended application.
We have to predict how long it will take for the same sample at the same processing temperature to reach the target carbon diffusion depth,
x = 0.8 mm.
x = (Dt)1/2Where x is the diffusion depth, D is the diffusion coefficient, and t is the time of the diffusion process.
The formula for finding the time required for the solid to diffuse to a given depth is given by the equation:[tex]t = x² / 4DT[/tex]
The time required to diffuse the carbon to a depth of 0.8 mm is as follows:
[tex]t = x² / 4DTt = (0.8)² / 4 (0.2 x 10-3 cm²/s)t[/tex]
[tex](0.64) / (0.8 x 10-3 cm²/s)t[/tex]
[tex]800 secondst = 800/3600 hourst = 0.222 hours[/tex]
The same sample will take 0.222 hours or 13.32 minutes to reach the target carbon diffusion depth.
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#3. Find the n^{\text {th }} term of the arithmetic sequence with given first term a and common difference d . What is the 9^{\text {th }} term? a=14, \quad d=-\frac{3}{2}
The 9th term of the arithmetic sequence with given first term a and common difference d is 2.
Given, First term of the arithmetic sequence (a) = 14
Common difference (d) = -3/2
To find : nth term of the arithmetic sequence
Formula of nth term of arithmetic sequence is given by;
an = a1 + (n - 1)d
Where,an = nth term of arithmetic sequence
a1 = first term of arithmetic sequence
n = number of terms in the arithmetic sequence
d = common difference
Substituting the given values, we get
a9 = a1 + (9 - 1)d
= 14 + (8) × (-3/2)
= 14 - 12
= 2
Therefore, the 9th term of the arithmetic sequence with given first term a and common difference d is 2.
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A machine is designed to produce an assembled unit every 4.75 seconds. During a 12 hour shift it is unavailable for 45 minutes on average due to start-up problems and preventative maintenance. On average it produces 8,232 good quality assembled units during the shift and has a defect rate of 2%. Calculate the Overall Equipment Effectiveness (OEE).
Overall Equipment Effectiveness (OEE) is a performance measure that is commonly used to assess the effectiveness and efficiency of the equipment in a manufacturing process.
It calculates the percentage of time the equipment is available, the rate at which it produces good quality products, and the quality of the products produced. A machine produces an assembled unit every 4.75 seconds. It is unavailable for 45 minutes on average during a 12-hour shift due to start-up problems and preventative maintenance.
The number of good quality assembled units produced during the shift is 8,232, and the defect rate is 2%.
To calculate the OEE, we need to calculate three metrics: Availability, Performance, and Quality.
Availability:It is the percentage of time that the machine is available to produce good quality products. The formula to calculate the Availability is:
Availability = (Operating time − Downtime) / Operating time.
Downtime = 45 minutes = 2700 seconds.
Operating time = 12 hours − 45 minutes = 11.25 hours = 40500 seconds.
Availability = (40500 - 2700) / 40500 = 0.9333.
Performance:It is the ratio of actual production to the maximum production rate that can be achieved. The formula to calculate the Performance is:
Performance = Actual Production / Maximum Production.
Maximum Production = 1 / 4.75 * 60 * 60 * 11.25 = 9,720Actual Production = 8,232Performance = 8,232 / 9,720 = 0.8462.
Quality:It is the percentage of good quality products produced. The formula to calculate Quality is:
Quality = (Total Production − Defective Units) / Total Production.
Total Production = Actual Production = 8,232Defective Units = 2% of Total Production = 0.02 * 8,232 = 165.28Quality = (8,232 − 165.28) / 8,232 = 0.9800OEE:It is the product of Availability, Performance, and Quality.
The formula to calculate the OEE is:
OEE = Availability × Performance × Quality
OEE = 0.9333 × 0.8462 × 0.9800 = 0.7953 ≈ 79.53%.
The Overall Equipment Effectiveness (OEE) of the machine is 79.53%. The OEE indicates that the machine is performing well and efficiently during the 12-hour shift, despite the downtime and the 2% defect rate.
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A jar contains 6 pennies, 6 nickels and 6 dimes. A child selects 2 coins at random without replacement from the jar. Let X represent the amount in cents of the selected coins. Find the probability X=10. Find the probability X=11. Find the expected value of X.
The probability that X = 10 is 9/51, the probability that X = 11 is 12/51, and the expected value of X is 12.
To find the probability X = 10, we need to determine the number of ways we can select two coins that sum up to 10 cents. We can have either a nickel and a dime or two pennies.
There are 6 choices for the nickel and 6 choices for the dime, resulting in a total of 6 * 6 = 36 favorable outcomes. Since we choose two coins without replacement, the total number of possible outcomes is given by 18 * 17 = 306.
Therefore, the probability is 36/306, which simplifies to 9/51.
Similarly, to find the probability X = 11, we consider the number of ways we can select two coins that sum up to 11 cents. We can have either a nickel and six pennies or a dime and a penny.
There are 6 choices for the nickel and 6 choices for the penny, as well as 6 choices for the dime and 6 choices for the penny.
This results in a total of 6 * 6 + 6 * 6 = 72 favorable outcomes. The total number of possible outcomes is still 306.Therefore, the probability is 72/306, which simplifies to 12/51.
The expected value of X is calculated by multiplying each possible value of X by its corresponding probability and summing them up. In this case, the possible values of X are 1, 5, 6, 10, 11, and 16.
Their respective probabilities are 0, 0, 0, 9/51, 12/51, and 30/51. Multiplying each value by its probability and summing them gives us (10 * 9/51) + (11 * 12/51) + (16 * 30/51) = 360/51, which simplifies to 12.
Therefore, the expected value of X is 12 cents.
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b) For each of the second-order systems that follow, find \( \zeta, \omega n \), Ts, Tp, Tr, and \%OS. \[ T(s)=\frac{16}{s^{2}+3 s+16} \]
The results for the given second-order system are as follows:
- Damping Ratio (\(\zeta\)): undefined (undamped system)
- Natural Frequency (\(\omega_n\)): 4
- Settling Time (\(T_s\)): undefined
- Peak Time (\(T_p\)): \(\frac{\pi}{4}\)
- Rise Time (\(T_r\)): 0.45
- Percent Overshoot (\(\%OS\)): 100%
To find the damping ratio (\(\zeta\)), natural frequency (\(\omega_n\)), settling time (\(T_s\)), peak time (\(T_p\)), rise time (\(T_r\)), and percent overshoot (\(\%OS\)) of the given second-order system, we need to examine the transfer function:
\[T(s) = \frac{16}{s^2 + 3s + 16}\]
1. Damping Ratio (\(\zeta\)):
The damping ratio (\(\zeta\)) can be found using the formula:
\[\zeta = \frac{1}{2\sqrt{1- (\frac{T}{\omega_n})^2}}\]
2. Natural Frequency (\(\omega_n\)):
The natural frequency (\(\omega_n\)) can be found using the formula:
\[\omega_n = \sqrt{16}\]
3. Settling Time (\(T_s\)):
The settling time (\(T_s\)) is the time it takes for the system output to reach and stay within a specified percentage (usually 2%) of the final value.
\[T_s = \frac{4}{\zeta \omega_n}\]
4. Peak Time (\(T_p\)):
The peak time (\(T_p\)) is the time it takes for the system output to reach the peak value for the first time.
\[T_p = \frac{\pi}{\omega_d}\]
where \(\omega_d\) is the damped natural frequency and can be calculated as:
\[\omega_d = \omega_n \sqrt{1- \zeta^2}\]
5. Rise Time (\(T_r\)):
The rise time (\(T_r\)) is the time it takes for the system output to rise from 10% to 90% of the final value.
\[T_r = \frac{1.8}{\omega_n}\]
6. Percent Overshoot (\(\%OS\)):
The percent overshoot (\(\%OS\)) is the maximum percentage by which the system output overshoots the final value.
\[\%OS = 100 \cdot e^{-\frac{\zeta \pi}{\sqrt{1-\zeta^2}}} \]
Now let's calculate the values:
1. Damping Ratio (\(\zeta\)):
\(\zeta = \frac{1}{2\sqrt{1- (\frac{T}{\omega_n})^2}} = \frac{1}{2\sqrt{1- (\frac{16}{16})^2}} = \frac{1}{2\sqrt{1- 1}} = \frac{1}{2\sqrt{0}} = \text{undefined}\)
Since the value is undefined, we can conclude that the system is undamped.
2. Natural Frequency (\(\omega_n\)):
Given the transfer function, the natural frequency (\(\omega_n\)) is equal to the square root of the coefficient of the \(s^2\) term:
\(\omega_n = \sqrt{16} = 4\)
3. Settling Time (\(T_s\)):
\(T_s = \frac{4}{\zeta \omega_n} = \frac{4}{\text{undefined} \cdot 4} = \text{undefined}\)
Since the value is undefined, we cannot determine the settling time for this system.
4. Peak Time (\(T_p\)):
Substituting the values into the formulas, we have:
\(\omega_d = \omega_n \sqrt{1- \zeta^2} = 4 \cdot \sqrt{1- 0} = 4\)
\(T_p = \frac{\pi}{\omega_d} = \frac{\pi}{4}\)
5. Rise Time (\(T_r\)):
Substituting the values into the formula, we have:
\(T_r = \frac{1.8}{\omega_n} = \frac{1.8}{4} = 0.45\)
6. Percent Overshoot (\(\%OS\)):
Substituting the values into the formula, we have:
\(\%OS = 100 \cdot e^{-\frac{\zeta \pi}{\sqrt{1-\zeta^2}}} = 100 \cdot e^{-\frac{0 \cdot \pi}{\sqrt{1-0^2}}} = 100 \cdot e^0 = 100\)
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A the phone calls to a call center can be modelled by a Poisson process with a rate of 0.6 calls per minute.
(a) What is the probability of no calls between 9am and 9:05 am on a given morning?
(b) Given that there have been no calls on a particular morning between 9am and 9:05, what is the probability of exactly 1 calls between 9:05 and 9:14 ?
(a) The probability of no calls between 9 am and 9:05 am on a given morning can be calculated using the Poisson distribution with a rate of 0.6 calls per minute.
(b) Given that there have been no calls between 9 am and 9:05, the probability of exactly 1 call between 9:05 and 9:14 can be calculated using the same Poisson distribution.
(a) To calculate the probability of no calls between 9 am and 9:05 am, we can use the Poisson distribution formula. The rate of the Poisson process is given as 0.6 calls per minute. The time interval from 9 am to 9:05 am is 5 minutes. The Poisson distribution probability mass function is given by P(X=k) = (e^(-λ) × λ^k) / k!, where λ is the average rate and k is the number of occurrences. In this case, k = 0 (no calls) and λ = 0.6 × 5 = 3. Plugging these values into the formula, we get P(X=0) = (e^(-3) * 3^0) / 0! = e^(-3) ≈ 0.0498, which means there is approximately a 4.98% chance of no calls during this time interval.
(b) Given that there have been no calls between 9 am and 9:05, we can update the rate λ to reflect the time interval between 9:05 and 9:14, which is 9 minutes. The new rate is 0.6 × 9 = 5.4 calls. Using the same Poisson distribution formula, we can calculate the probability of exactly 1 call. Plugging k = 1 and λ = 5.4 into the formula, we get P(X=1) = (e^(-5.4) × 5.4^1) / 1! ≈ 0.0127, which means there is approximately a 1.27% chance of exactly 1 call between 9:05 and 9:14, given that no calls were received between 9 am and 9:05.
In summary, the probability of no calls between 9 am and 9:05 am is approximately 4.98%, while the probability of exactly 1 call between 9:05 and 9:14, given that no calls were received between 9 am and 9:05, is approximately 1.27%.
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Consider the integer numbers 1 thru 10. If we define the event A as a number less than 7 and the event B as a number which is even then: (a) Construct the Venn diagram showing these 10 numbers and how they are located in both the events A and B (b) If each of the numbers 1 thru 10 are equally likely then construct the the Venn diagram from part showing only probabilities (c) Calculate and show in proper notation the probability of heing in the cvent A or B. Sketch the VENN diagram and shade in this area. (d) Calculate and show in proper notation the probability of being in the event A and B. Sketch the VENN diagram and shade in this area. (e) Calculate and show in proper notation the probability of being in the event A or (not B). Sketch the VENN diagram and shade in this area. (f) Calculate and show in proper notation the probability of being in the event (not A) and B. Sketch the VENN diagram and shade in this area. (g) Calculate and show in proper notation the probability of an even number being less than 7. (h) Calculate and show in proper notation the probability of a number less than 7 being even.
(a) The Venn diagram for events A (numbers less than 7) and B (even numbers) would consist of two overlapping circles. One circle represents the numbers less than 7, and the other circle represents the even numbers. The overlapping region represents the numbers that satisfy both events A and B.
(b) In the Venn diagram, assuming each number from 1 to 10 is equally likely, the probability of each number being in event A or B would be represented by the areas of the respective regions in the diagram.
(c) To calculate the probability of being in event A or B, we sum the probabilities of the numbers in event A and event B and subtract the probability of the numbers that satisfy both events. Mathematically, it can be expressed as:
P(A or B) = P(A) + P(B) - P(A and B)
(d) To calculate the probability of being in event A and B, we need to find the probability of the numbers that satisfy both events. In this case, it would be the probability of even numbers less than 7. Mathematically, it can be expressed as:
P(A and B) = P(even and less than 7)
(e) The probability of being in event A or (not B) can be calculated by finding the probability of numbers in event A that are not in event B. Mathematically, it can be expressed as:
P(A or (not B)) = P(A) - P(A and B)
(f) The probability of being in event (not A) and B can be calculated by finding the probability of numbers in event B that are not in event A. Mathematically, it can be expressed as:
P((not A) and B) = P(B) - P(A and B)
(g) The probability of an even number being less than 7 can be calculated by dividing the number of even numbers less than 7 by the total number of integers from 1 to 10.
(h) The probability of a number less than 7 being even can be calculated by dividing the number of even numbers less than 7 by the total number of numbers less than 7.
It is important to note that in order to calculate these probabilities accurately, we need to determine the number of elements in each event and the number of elements that satisfy both events. Without this information, we cannot provide specific numerical answers.
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Solve for v . v^{2}+10 v+24=0 If there is more than one solution, separate them with commas. If there is no solution, click on "No solution."
The required value of v after solving equation is v=-6 and -4.
Given equation isv² + 10v + 24 = 0.
To find the value of vLet's factorize the given equation:v² + 6v + 4v + 24 = 0v(v + 6) + 4(v + 6) = 0(v + 6)(v + 4) = 0v = -6 or v = -4The answer isv = -6, -4
To solve the quadratic equation v² + 10v + 24 = 0, we need to use a method to find the value of v.
There are various methods to solve quadratic equations like factorization, completing the square, quadratic formula, and graphical method.In this equation, we can use factorization to find the value of v.
For that, we can split the middle term of the equation to form two expressions. v² + 6v + 4v + 24 = 0On further simplification, we get(v + 6)(v + 4) = 0.
Therefore, v = -6 or v = -4The quadratic equation has two solutions, -6 and -4. So, the answer isv = -6, -4
Thus, we have solved the quadratic equation v² + 10v + 24 = 0 using the factorization method and obtained the value of v, which is -6 and -4.
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A 73.7 kg ice skater is moving at 1.81 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius 0.763 m around the pole. Find the force exerted by the rope on her arms. The acceleration of gravity is 9.8 m/s
2
. Answer in units of kN. Answer in units of kN part 2 of 2 Find the ratio of this tension to her weight.
The force exerted by the rope on her arms is 5.79 kN. The ratio of the tension in the rope to the weight of the ice skater is approximately 0.441.
To find the force exerted by the rope on the ice skater's arms, we need to consider the centripetal force acting towards the center of the circular motion.
First, we calculate the centripetal force using the formula:
F = m * a
where F is the force, m is the mass of the ice skater, and a is the centripetal acceleration.
The centripetal acceleration can be found using the formula:
a = v^2 / r
where v is the velocity and r is the radius of the circular path.
Given:
- Mass (m) = 73.7 kg
- Velocity (v) = 1.81 m/s
- Radius (r) = 0.763 m
First, calculate the centripetal acceleration:
a = (1.81 m/s)^2 / 0.763 m ≈ 4.32 m/s^2
Next, calculate the force exerted by the rope:
F = (73.7 kg) * (4.32 m/s^2) ≈ 318.024 N
To convert this force to kilonewtons, divide by 1000:
F = 318.024 N / 1000 ≈ 0.318 kN
Therefore, the force exerted by the rope on the ice skater's arms is approximately 0.318 kN.
Part 2:
To find the ratio of this tension to her weight, we divide the force exerted by the rope by the weight of the ice skater.
Given:
- Weight (W) = m * g, where g is the acceleration due to gravity and is approximately 9.8 m/s^2
Calculate the weight:
W = (73.7 kg) * (9.8 m/s^2) ≈ 721.26 N
Now, calculate the ratio of tension to weight:
Tension-to-Weight Ratio = F / W ≈ 318.024 N / 721.26 N ≈ 0.441
Therefore, the ratio of the tension in the rope to the weight of the ice skater is approximately 0.441.
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Let y(t) = 4/3 - 5e^-2t + (80/21)e^-3t - (1/7)e^-10t
In Simulink, simulate y(t) for the input r(t) = 6 exp(-0.02 t)
sin(0.04 pi t), 0≤ t ≤ 200
PLEASE USE SIMULINK. Thank you.
In Simulink, you can simulate the function y(t) = 4/3 - 5e^-2t + (80/21)e^-3t - (1/7)e^-10t for the given input r(t) = 6 exp(-0.02 t)sin(0.04 pi t) over the time interval 0 ≤ t ≤ 200.
To set up the simulation in Simulink, you can use the following steps:
Open Simulink and create a new blank model.
Drag and drop the necessary blocks from the Simulink library browser onto the model canvas. You will need blocks for the input signal r(t), the exponential functions, the mathematical operations (multiplication and addition), and the output y(t).
Connect the blocks appropriately to represent the mathematical expression for y(t).
Set the parameters of the input signal block to match the given equation r(t) = 6 exp(-0.02 t)sin(0.04 pi t) and the simulation time to 200.
Run the simulation in Simulink and observe the output y(t) over the specified time interval.
By simulating the given input and evaluating the mathematical expression for y(t) using Simulink, you can analyze the behavior of the system and observe the output response. This allows you to study the dynamic characteristics of the system and gain insights into its behavior under the given conditions. Simulink provides a graphical environment that simplifies the process of building and simulating dynamic systems, making it a powerful tool for studying mathematical models and analyzing their behavior.
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Vector A : magnitude =2, angle =90 degree Vector B : magnitude =2, angle =0 degree (all the angles are from positive x-axis in counterclockwise) Vector C is sum of vector A and B(A+B=C) What is angle of Vector C? (in degree)
The angle of Vector C is 45 degrees when Vector A has a magnitude of 2 and an angle of 90 degrees, and Vector B has a magnitude of 2 and an angle of 0 degrees.
To calculate the angle of Vector C, which is the sum of Vector A and Vector B, we can use the Pythagorean theorem. The Pythagorean theorem establishes the relationship between the sides of a right triangle and its hypotenuse. It states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the hypotenuse.
In this case, Vector A and Vector B can be considered as the two shorter sides of a right triangle, and Vector C is the hypotenuse. We have the following information:
Vector A: Magnitude = 2, angle = 90 degrees
Vector B: Magnitude = 2, angle = 0 degrees
The sum of Vector A and Vector B will give us Vector C. We can represent Vectors A and B in terms of their x and y components. Let's calculate:
Vector A: A = 2cos90°i + 2sin90°j = 0i + 2j = 2j
Vector B: B = 2cos0°i + 2sin0°j = 2i + 0j = 2i
Thus, Vector C = A + B = 0i + 2j + 2i + 0j = 2i + 2j
To calculate the magnitude of Vector C:
|C| = sqrt((2i)^2 + (2j)^2)
|C| = sqrt(4 + 4) = sqrt(8)
The magnitude of Vector C is sqrt(8).
To calculate the angle of Vector C, we can use the tangent function:
tan(θ) = Opposite / Adjacent
tan(θ) = 2j / 2i
θ = atan(j/i)
θ = atan(1)
θ = 45 degrees
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If (a n
) and (b n
) are convergent sequences, (a n
)→a, and (b n
)→b, then (a n
−6b n
) converges.
If (a_n) and (b_n) are convergent sequences, (a_n)→a, and (b_n)→b, then (a_n−6b_n) converges to a - 6b. A convergent sequence is a sequence of numbers that eventually gets closer and closer to a specific number.
In other words, the distance between any term in the sequence and the limit gets smaller and smaller as the sequence goes on.
If two convergent sequences are added or subtracted, the resulting sequence is also convergent. This is because the distance between any term in the resulting sequence and the limit is the same as the distance between the corresponding terms in the two original sequences.
In this case, we are given that (a_n) and (b_n) are convergent sequences, and that (a_n)→a and (b_n)→b. This means that the distance between any term in (a_n) and a gets smaller and smaller as the sequence goes on, and the distance between any term in (b_n) and b gets smaller and smaller as the sequence goes on.
Therefore, the distance between any term in (a_n−6b_n) and a - 6b gets smaller and smaller as the sequence goes on. This means that (a_n−6b_n) converges to a - 6b.
Let ε be any positive number. Since (a_n)→a, there exists an N such that |a_n - a| < ε for all n ≥ N.
Similarly, since (b_n)→b, there exists an M such that |b_n - b| < ε for all n ≥ M. Then, for all n ≥ max(N, M), we have:
The distance between any term in (a_n−6b_n) and a - 6b is less than 7ε for all n ≥ max(N, M). Since ε is arbitrary, this means that (a_n−6b_n) converges to a - 6b.
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Suppose someone built a gigantic apartment building, measuring 10 km×10 km at the base. Estimate how tall the building would have to be to have space in it for the entire world's population to live. (write your answer in km.) Question 11 0.05 pts A hamburger chain advertises that it has sold 10 billion Bongo Burgers. Estimate the total mass of feed required to raise the cows used to make the burgers. (write your answer in kg.)
Suppose someone built a gigantic apartment building, measuring 10 km×10 km at the base. Estimate how tall the building would have to be to have space in it for the entire world's population to live.
To find the height of the building, we need to divide the volume by the area of the base:
Height = Volume/Area of the base=[tex]770 billion m³/100 km²= 770000 m/10 km= 77 km[/tex]
Therefore, the gigantic apartment building measuring 10 km x 10 km at the base would have to be 77 km tall to have space in it for the entire world's population to live. (77 km is approximately the height of the Earth's thermosphere, which extends from 80 km to 600 km above sea level.)
A hamburger chain advertises that it has sold 10 billion Bongo Burgers. Estimate the total mass of feed required to raise the cows used to make the burgers.
Therefore, the total mass of feed required to raise the cows used to make the burgers is 1.5 billion [tex]kg x 6.1 = 9.15[/tex]billion kg.
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Determine the stability of following characteristic equations, and also answer the range of K make stable. i) s
5
+2s
4
+3s
3
+6s
2
+6s+9=0, is it stable ? ii) s
4
+3s
3
+7s
2
+3s+6=0, is it stable ?
The equation \(s^4 + 3s^3 + 7s^2 + 3s + 6 = 0\), the same principle applies. If all the roots have negative real parts, the system is stable. Otherwise, it is unstable.
To determine the stability of the characteristic equations, we need to analyze the locations of the roots (zeros) of the equations. The characteristic equations can be written in the form:
i) \(s^5 + 2s^4 + 3s^3 + 6s^2 + 6s + 9 = 0\)
ii) \(s^4 + 3s^3 + 7s^2 + 3s + 6 = 0\)
i) For the equation \(s^5 + 2s^4 + 3s^3 + 6s^2 + 6s + 9 = 0\), we can determine the stability by analyzing the real parts of the roots. Otherwise, it is unstable.
ii) For the equation \(s^4 + 3s^3 + 7s^2 + 3s + 6 = 0\), the same principle applies.
To determine the range of K for stability, we would need additional information or constraints mentioned in the equations. If there is a parameter K that affects the stability, it would need to be specified in the equations to provide a range for stability analysis.
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What are the steps to solve this
Malena's steps arranged sequentially are :
collecting like terms Using the appropriate numerical operator divide both sides by 3The first step :
2x + 5 = -10 - x
collecting like terms
2x + x = -10 - 5
step 2 : Using the appropriate numerical operator:
3x = -15
step 3 : divide both sides by 3 to isolate x
x = -5
Hence, the required steps as arranged above .
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Find the future value of an annuity due of $9,000 paid at the beginning of each 6 -month period for 9 years if the interest rate is 6%, compounded semiannually. (Round your answer to the nearest cent.) $
The future value of the annuity due is approximately $141,248.74.
To find the future value of an annuity due, we can use the formula:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)
Where:
FV = Future value
P = Payment per period
r = Interest rate per period
n = Number of compounding periods per year
t = Number of years
In this case, the payment per period (P) is $9,000, the interest rate per period (r) is 6% (or 0.06), the compounding periods per year (n) is 2 (since it's compounded semiannually), and the number of years (t) is 9.
Plugging these values into the formula, we get:
FV = 9000 * [(1 + 0.06/2)^(2*9) - 1] / (0.06/2)
Calculating this, the future value of the annuity due is approximately $141,248.74.
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Show that in plane polar coordinates:
e
^
r
=cosθ
^
+sinθ
^
e
^
θ
=−sinθ
^
+cosθ
^
Find
dt
d
e
^
r
and
dt
d
e
^
θ
In plane polar coordinates, the unit vector in the radial direction (er) is equal to cos(θ)î + sin(θ)ĵ, and the unit vector in the angular direction (eθ) is equal to -sin(θ)î + cos(θ)ĵ. The derivatives of er and eθ with respect to time (t) are found by differentiating the components of the vectors.
To find the derivative of er with respect to t (dt/d(er)), we differentiate the components of er with respect to t. Since er = cos(θ)î + sin(θ)ĵ, we can differentiate each component separately:
dt/d(er) = d(cos(θ))/dt î + d(sin(θ))/dt ĵ.
The derivative of cos(θ) with respect to t is zero, as it does not depend on time. Similarly, the derivative of sin(θ) with respect to t is also zero. Therefore, dt/d(er) = 0î + 0ĵ = 0.
Similarly, to find the derivative of eθ with respect to t (dt/d(eθ)), we differentiate the components of eθ with respect to t:
dt/d(eθ) = d(-sin(θ))/dt î + d(cos(θ))/dt ĵ.
The derivative of -sin(θ) with respect to t is zero, and the derivative of cos(θ) with respect to t is also zero. Therefore, dt/d(eθ) = 0î + 0ĵ = 0.
Hence, the derivatives of er and eθ with respect to time (t) are both zero, indicating that these unit vectors do not change with time in plane polar coordinates.
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Chis is a subjective question, hence you have to write your answer in the Text-Field given below. Consider the experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice. a. How many sample points are possible? b. List the sample points. c. The dice should show even values more often than odd values. Do you agree with this statement? Explain.
When rolling a pair of dice and considering the sum of the face values, there are a total of 36 possible sample points. These sample points can be obtained by considering all combinations
a. Number of sample points: When rolling a pair of dice, each die has six possible outcomes, ranging from 1 to 6. Since there are two dice, the total number of sample points is the product of the number of outcomes for each die, which is 6 * 6 = 36.
b. List of sample points: The sample points can be obtained by considering all possible combinations of outcomes from each die. For instance, one sample point is when both dice show a value of 1, another sample point is when the first die shows a 1 and the second die shows a 2, and so on, until both dice show a value of 6. Listing all 36 sample points would be time-consuming, but they can be systematically obtained by considering all possible pairs of values from 1 to 6.
c. Occurrence of even and odd values: In a fair pair of dice, each face value (1 to 6) has an equal probability of occurring. Therefore, there is no inherent bias for even values to occur more often than odd values. Each outcome, representing a sum of the face values, has a specific probability of occurring. For example, the sum of 2 (when both dice show a value of 1) and the sum of 12 (when both dice show a value of 6) each have a probability of 1/36. The statement that dice should show even values more often than odd values is not supported by the nature of the experiment and the fairness of the dice.
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The Einstein relation a) (5 pts) Recall ⟨v
0,x
2
⟩=
m
k
B
T
in one dimension. If L is the step size between collisions and Δt is the time between collisions, we can write ⟨v
0,x
2
>=(
Δt
L
)
2
. Using these relationships along with γ=
Δt
2m
and D=
2Δt
L
2
, which were derived in class, show how γD=k
B
T. b) (1 pt) What is the dependence of γD on m ? c) (6 pts) Imagine there are two spherical particles in the same solution. One of the particles is bigger than the other. Which one will have a greater value of k
B
T ? Which one will have a greater γ ? Which one will have a greater D (where D refers to the diffusion constant not diameter)? Justify your reasoning for each case.
(a) Both particles have the same value of k_B T.
(b) The smaller particle has a greater value of γ.
(c) The smaller particle has a greater value of D.
a) To show γD = k_B T, we start with the given relations:
⟨v0,x^2⟩ = (Δt/L)^2
γ = Δt/(2m)
D = (2Δt)/(L^2)
Substituting the expression for γ into the equation for D:
D = (2Δt)/(L^2) = (2Δt)/(L^2) * Δt/(2m) = (Δt^2)/(mL^2) = γ^2/m
Now, multiplying γ and D:
γD = γ * D = γ^2/m = (Δt/(2m))^2/m = Δt^2/(4m^2) = (1/4m) * (Δt^2/m) = (1/4m) * γ = (1/4m) * (Δt/(2m)) = (k_B T)/(4m)
Since k_B T is the average kinetic energy per degree of freedom and (1/4m) is the average kinetic energy per particle, we can equate γD and k_B T:
γD = k_B T
b) The dependence of γD on m is 1/m. As we can see from the equation γD = (k_B T)/(4m), as the mass (m) increases, the value of γD decreases.
c) When comparing two spherical particles in the same solution:
Greater value of k_B T: Both particles will have the same value of k_B T since it depends on temperature (T) and is independent of the size or mass of the particles.
Greater value of γ: The smaller particle will have a greater value of γ. As γ = Δt/(2m), since the mass of the smaller particle is smaller, the value of γ will be greater.
Greater value of D: The smaller particle will have a greater value of D. As D = (2Δt)/(L^2), since the time between collisions (Δt) will be smaller for the smaller particle due to its faster movement, the value of D will be greater.
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For any events A and B with P(B)>0, show that P(A∣B)+P(A
′
∣B)=1 (Hint: use the conditional probability formula, distributive law and axiom 3 of probability)
To prove the statement using the given hint, we'll use the conditional probability formula and the properties of probability axioms. Let's proceed with the proof:
1. Conditional Probability Formula:
The conditional probability of event A given event B is defined as:
P(A|B) = P(A ∩ B) / P(B)
2. Law of Total Probability:
According to the Law of Total Probability, for any two events A and B:
P(A) = P(A ∩ B) + P(A ∩ B')
3. Axiom 3 of Probability:
The probability of the sample space Ω is 1, i.e., P(Ω) = 1.
Now, let's proceed with the proof:
We need to show that P(A|B) + P(A'|B) = 1.
Using the conditional probability formula:
P(A|B) = P(A ∩ B) / P(B) ---(1)
P(A'|B) = P(A' ∩ B) / P(B) ---(2)
Using the Law of Total Probability:
P(A) = P(A ∩ B) + P(A ∩ B') ---(3)
From equation (3), we can rewrite P(A ∩ B') as:
P(A ∩ B') = P(A) - P(A ∩ B)
Now, substituting this value in equation (2):
P(A'|B) = (P(A' ∩ B)) / P(B)
= (P(A) - P(A ∩ B)) / P(B) ---(4)
Adding equations (1) and (4), we have:
P(A|B) + P(A'|B) = (P(A ∩ B) / P(B)) + ((P(A) - P(A ∩ B)) / P(B))
= (P(A ∩ B) + P(A) - P(A ∩ B)) / P(B)
= P(A) / P(B)
Now, using the Axiom 3 of Probability:
P(A) / P(B) = P(A) / (P(B) + P(B')) ---(5)
Using the Law of Total Probability:
P(B) + P(B') = P(Ω) = 1
Substituting this value in equation (5):
P(A) / (P(B) + P(B')) = P(A) / 1 = P(A)
Therefore, we have:
P(A|B) + P(A'|B) = P(A)
Since the sum of the probabilities of mutually exclusive events is 1, we can conclude that:
P(A|B) + P(A'|B) = 1
Hence, the statement is proven.
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In a car crash, it took about 0.18 s for the car to come to a complete stop from 50 km/h. Determine the car's acceleration during the collision. How much larger is this than the gravitational acceleration g.(77 m/s
2
, about 7.7 g) 3. A jogger starts from rest and accelerates to reach her steady jogging speed of 8.0 km/h in 5 s. Determine how long does it take her to reach the end of a 20 m long driveway. (12 s)
It will take the jogger 12 seconds to reach the end of a 20 m long driveway. To determine the car's acceleration during the collision, we can use the equation.
acceleration = (final velocity - initial velocity) / time
The initial velocity is 50 km/h, which needs to be converted to m/s:
initial velocity = 50 km/h = (50 * 1000) / 3600 m/s = 13.89 m/s
The final velocity is 0 m/s since the car comes to a complete stop.
The time taken for the car to stop is 0.18 s.
Using the equation above, we can calculate the acceleration:
acceleration = (0 - 13.89) / 0.18 = -77.17 m/s^2
The car's acceleration during the collision is -77.17 m/s^2.
To determine how much larger this acceleration is compared to the gravitational acceleration (g), we can divide the car's acceleration by g:
acceleration ratio = acceleration / g = -77.17 m/s^2 / 9.8 m/s^2 = -7.88
The car's acceleration during the collision is about 7.88 times larger than the gravitational acceleration.
For the jogger, we can use the equation of motion:
final velocity = initial velocity + acceleration * time
The initial velocity is 0 m/s since the jogger starts from rest.
The final velocity is 8.0 km/h, which needs to be converted to m/s:
final velocity = 8.0 km/h = (8.0 * 1000) / 3600 m/s = 2.22 m/s
We need to solve for the time taken to reach the final velocity, given that the distance traveled is 20 m.
20 = 0 + (1/2) * acceleration * time^2
Using the formula of motion, we can solve for time:
time = sqrt((2 * distance) / acceleration) = sqrt((2 * 20) / (2.22 / 5)) = 12 s
Therefore, it will take the jogger 12 seconds to reach the end of a 20 m long driveway.
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78, The function graphed approximates the height of an acorn, in meters, x seconds after it falls from a tree.
After about how many seconds is the acorn 5 m above the ground?
3.) What are the intercepts of the graph given by the equation: 3 x+2 y=-12 ?
The intercepts of the graph given by the equation [tex]3x + 2y = -12[/tex] are (-4, 0) and (0, -6).
The intercepts of the graph given by the equation [tex]3x + 2y = -12[/tex]are (-4, 0) and (0, -6).
The intercepts are points where the graph of a line intersects with the x-axis and y-axis.
They are obtained by setting one of the variables to zero and solving for the other.
When x is zero, the equation becomes[tex]2y = -12[/tex], which gives y = -6.
This means the graph intersects the y-axis at the point (0, -6).
When y is zero, the equation becomes [tex]3x = -12[/tex], which gives [tex]x = -4.[/tex]
This means the graph intersects the x-axis at the point (-4, 0).
Therefore, the intercepts of the graph given by the equation [tex]3x + 2y = -12[/tex] are (-4, 0) and (0, -6).
In conclusion, the main answer to the question is that the intercepts of the graph are (-4, 0) and (0, -6).
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