The area under the curve is found to be 8 square units for the given function of f(x) = x² + 1.
Given function is f(x) = x² + 1 over the interval [0, 2]. We have to find the following:
Δx, x_k, x_k* as the left endpoint or right endpoint, f(x_k*) Δr, and the area under the curve.
Here, a is the left endpoint of the interval and b is the right endpoint of the interval.
So, a = 0 and b = 2.
(a)Δx = Δx = (b - a)/n, where n is the number of sub-intervals.
Substituting a = 0, b = 2, and n = 2,
Δx = (2 - 0)/2
= 1.
Thus, Δx = 1.
(b)x_k = a + k Δx,
where k = 0, 1, 2, ..., n - 1.
For k = 0,
x_0 = 0 + 0 × 1
= 0.
For k = 1,
x_1 = 0 + 1 × 1
= 1.
For k = 2,
x_2 = 0 + 2 × 1
= 2.
(c) For the left endpoint,
x_k* = x_k
= x₀, x₁, x₂, ...
For the right endpoint,
x_k* = x_k + 1
= x₁, x₂, x₃, ...
Since we have to find x_k* as the left endpoint or right endpoint, we take the left endpoint.
For k = 0,
x_k* = x₀
= 0.
For k = 1,
x_k* = x₁
= 1.
For k = 2,
x_k* = x₂
= 2.
(d)We have to find f(x_k*) Δr.
f(x) = x² + 1.
Putting x = x₀,
f(x₀) = x₀² + 1
= 0 + 1
= 1.
f(x) = x² + 1.
Putting x = x₁,
f(x₁) = x₁² + 1
= 1² + 1
= 2.
f(x) = x² + 1.
Putting x = x₂,
f(x₂) = x₂² + 1
= 2² + 1
= 5.
Now, Δr = Δx = 1.
So, for k = 0,
f(x_k*) Δr = f(x₀) Δr
= 1 × 1
= 1.
For k = 1,
f(x_k*) Δr = f(x₁) Δr
= 2 × 1
= 2.
For k = 2, f(x_k*) Δr
= f(x₂) Δr
= 5 × 1
= 5.
(e)Now, we have to find the area under the curve.
The formula for the area under the curve using the left endpoint is given by:
Σf(x_k*) Δx, where k = 0, 1, 2, ..., n - 1.
Putting n = 2,
Σf(x_k*) Δx = f(x₀) Δx + f(x₁) Δx + f(x₂) Δx
= 1 × 1 + 2 × 1 + 5 × 1
= 1 + 2 + 5
= 8.
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∣ψ
1
=5∣1⟩−3i∣2⟩+2∣3⟩ ∣ψ
2
=1∣1⟩−5i∣2⟩+x∣3⟩
The answer is "No possible value for x".
It seems like you have provided two quantum states, denoted as |ψ1⟩ and |ψ2⟩. |ψ1⟩ and |ψ2⟩ are represented as linear combinations of the basis states |1⟩, |2⟩, and |3⟩. The coefficients in front of each basis state represent the probability amplitudes.
|ψ1⟩ = 5|1⟩ - 3i|2⟩ + 2|3⟩
|ψ2⟩ = 1|1⟩ - 5i|2⟩ + x|3⟩
In these expressions, |1⟩, |2⟩, and |3⟩ are basis states, and the coefficients 5, -3i, 2, 1, -5i, and x are probability amplitudes. The probability amplitudes determine the probabilities of measuring the system in each of the corresponding basis states.
Therefore, the answer is "No possible value for x".
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Find the exact value of each of the remaining trigonometric functions of θ. cosθ=− 25/24,θ in quadrant III sinθ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) tanθ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) cotθ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) secθ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) cscθ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Hence, sin θ = -7/24, tan θ = 7/25, cot θ = 25/7, sec θ = -24/25 and csc θ = -24/7.
Given cos θ = -25/24 and θ lies in quadrant III.
Trigonometric ratios of θ can be found as follows:
sin θ = sqrt(1-cos²θ)
sin θ = sqrt(1-(-25/24)²)
sin θ = sqrt(1-625/576)
sin θ = sqrt((576-625)/576)
sin θ = sqrt(-49/576)
As θ lies in quadrant III, sin θ is negative.
So,
sin θ = -sqrt(49/576)
sin θ = -7/24
tan θ = sin θ/cos θ
tan θ = (-7/24)/(-25/24)
tan θ = 7/25
cot θ = cos θ/sin θ
cot θ = (-25/24)/(-7/24)
cot θ = 25/7
sec θ = 1/cos θ
sec θ = -24/25
csc θ = 1/sin θ
csc θ = -24/7.
The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). In geometry, trigonometry is a branch of mathematics that deals with the sides and angles of a right-angled triangle. Therefore, trig ratios are evaluated with respect to sides and angles.
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Let G(u, v) = (Tu + v, 24u + 13u) be a map from the w.plane to the xy-plane. Find the image of the line through the points (u, v) = (1, 1) and (1,0) = (1, -1) under G in slope-intercept form. (Express numbers in exact form. Use symbolic notation and fractions where needed.) equation:
The equation of the image of the line through the points (u, v) = (1, 1) and (1,0) = (1, -1) under G in slope-intercept form is y = 24x + 13 - 24T.
Let us begin by finding the slope of the line through (u, v) = (1, 1) and (1,0) = (1, -1).
The slope of the line passing through two points (x1,y1) and (x2,y2) is given by
Slope = (y2-y1)/(x2-x1)
So the slope of the line through (1,1) and (1,0) is given by
(0 - 1) / (1 - 1) = -1/0, which is undefined.
Now we will get the equation of the line passing through (1,1) and (1,0).
The slope-intercept form of a line is given by y = mx + b where m is the slope of the line and b is the y-intercept.
So the equation of the line through (1,1) and (1,0) is x = 1.
Given that, G(u, v) = (Tu + v, 24u + 13u) is a map from the w-plane to the xy-plane.
The image of the line through the points (u, v) = (1, 1) and (1,0) = (1, -1) under G in slope-intercept form is to be determined.
To get the image of the line through the points (u, v) = (1, 1) and (1,0) = (1, -1),
we need to find the image of these points under G:
G(1, 1) = (T + 1, 37)and G(1, -1) = (T - 1, -11)
The slope of the line passing through the two points (T + 1, 37) and (T - 1, -11) is given by:
Slope = (-11 - 37) / (T - 1 - (T + 1))
= -48/-2
= 24
Therefore, the equation of the line passing through the two points (T + 1, 37) and (T - 1, -11) in slope-intercept form is given by:y = 24x + c where c is the y-intercept.
We can get the value of c by substituting the coordinates of one of the points (T + 1, 37) or (T - 1, -11):
37 = 24(T + 1) + c
c = 37 - 24(T + 1)
= 13 - 24T
Therefore, the equation of the line in slope-intercept form is given by:y = 24x + 13 - 24T
The equation of the image of the line through the points (u, v) = (1, 1) and (1,0) = (1, -1) under G in slope-intercept form is y = 24x + 13 - 24T.
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The X(bar) (bar) for a given process with 3 samples is 50. The
standard deviation for each sample follow
Sample 1: 5
Sample 2: 3
Sample 3: 7
A3 = 1.954
What is the upper control limit and lower contro
The upper control limit (UCL) and lower control limit (LCL) for the given process can be calculated using the formula: UCL = X(bar) + A3 × σ and LCL = X(bar) - A3 * σ. Based on the provided data, with X(bar) = 50 and the standard deviations of the three samples given as 5, 3, and 7, the values of UCL and LCL can be determined.
To calculate the UCL and LCL, we use the formula UCL = X(bar) + A3 × σ and LCL = X(bar) - A3 × σ. Here, X(bar) represents the sample mean, A3 is a constant factor (given as 1.954), and σ denotes the standard deviation. Given X(bar) = 50 and the standard deviations for the three samples as 5, 3, and 7, we can calculate the overall standard deviation by taking the average of the individual sample standard deviations. Thus, σ = (5 + 3 + 7) / 3 = 5. Using these values in the formulas, we find UCL = 50 + 1.954 × 5 = 59.77 and LCL = 50 - 1.954 × 5 = 40.23. Therefore, the upper control limit is approximately 59.77 and the lower control limit is approximately 40.23 for the given process.
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You won the state lottery this morning and you have two options to claim your winnings of $45M : You can ask for an annuity of 25 years (equal payments at the end of each year), or you can get a lump-sum of $26.5M. If you believe that the proper discount rate for this cash flow stream is 4.7%, which option do you choose and how much more do you get in today's dollars?
Choosing the lump-sum option of $26.5M would result in receiving approximately $2.93M more in today's dollars compared to the annuity option.
To determine which option is more advantageous, we need to compare the present value of the annuity payments with the lump-sum amount. The present value is calculated by discounting future cash flows at the appropriate discount rate.
For the annuity option, we have equal payments for 25 years. Using the discount rate of 4.7%, we calculate the present value of the annuity payments.For the lump-sum option, we have a single payment of $26.5M.
By discounting the annuity payments and summing them up, we find that the present value of the annuity is lower than the lump-sum amount. The difference between the present value of the annuity and the lump-sum is approximately $2.93M.
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Determine whether the following equation is separable. If so, solve the given initial value problem. 2yy′(t)=3t^2, y(0)=4
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution to the initial value problem is y(t)=__________ (Type an exact answer.)
B. The equation is not separable.
The given differential equation is 2yy′(t) = 3t². We need to find out if the given equation is separable or not.Separable equations are the differential equations in which the variables can be separated on different sides of the equation, so that the equation can be written in the form of `dy/dx = f(x)g(y)`.In the given equation, we can write the equation as `y' = (3t²)/(2y)`.
This is not a separable equation as we can't separate the variables in such a way that we have `dy/y = f(t)dt`. Hence, we cannot solve the equation using separation of variables method. The equation is not separable. Now we use a different method to solve the equation.
To solve the given initial value problem, we use the substitution method which is also known as homogeneous equation method. We can write the equation as `y' = (3t²)/(2y)`.Multiplying the above equation with y, we get `y * y' = (3t²)/2`.Substituting `u = y²`, we get `du/dt = 2y * y'`.
Substituting the values of `y'` and `y * y'` in the above equation, we get `du/dt = 3t²/u`.Now, we have a separable equation, which we can write as: `du/u = 3t²dt`. Integrating both sides of the equation, we get `ln|u| = t³ + C`.Here, C is the constant of integration.
Exponentiating both sides, we get `u = e^(t³ + C)`.Substituting the value of u, we get `y² = e^(t³ + C)`.Taking the square root, we get `y = ±√e^(t³ + C)`.Substituting the initial condition `y(0) = 4`, we get `y = ±2e^(t³)/√e^C`
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Let X
1
,⋯,X
n
be a random sample from a distribution with the pdf given by f
θ,λ
(x)=
θ
1
exp(−
θ
x−λ
) if x≥λ, otherwise f
θ,λ
(x)=0, where θ>0. Find the MLE's of θ and λ. Start by writing the likelihood function and note the constraint involving λ.
To find the maximum likelihood estimators (MLEs) of θ and λ, we start by writing the likelihood function for the given random sample. The constraint involving λ is that it should be less than or equal to the minimum value of the sample.
The likelihood function L(θ,λ) is obtained by taking the product of the individual probabilities for each observation in the sample. Since the random sample follows a given probability density function (pdf), we can write the likelihood function as:
L(θ,λ) = ∏[θe^(-θ(x_i - λ))] if x_i ≥ λ, otherwise L(θ,λ) = 0
To find the MLEs of θ and λ, we maximize this likelihood function. Taking the natural logarithm of the likelihood function (ln L(θ,λ)) simplifies the maximization process.
Since ln is a monotonically increasing function, maximizing ln L(θ,λ) is equivalent to maximizing L(θ,λ). Hence, we consider ln L(θ,λ) for simplicity. Taking the natural logarithm, we have:
ln L(θ,λ) = ∑[ln(θ) - θ(x_i - λ)]
To find the MLEs, we differentiate ln L(θ,λ) with respect to θ and λ, and set the derivatives equal to zero. Solving these equations will give us the MLEs. However, there is a constraint involving λ: it should be less than or equal to the minimum value of the sample. This constraint needs to be taken into account when finding the MLE for λ.
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Which of the following is not necessary to draw a box plot? a. Mean b. Median c. O
1
d. Q
3
14. Use the empirical rule to answer the following question. The mean of a set of data is 50 , the median is 55,Q
1
is 25 , and Q
3
is 85 . The highest and lowest values are 10 and 90 . The variance is 25 . Between what two values would about 95% of the data fall? a. 40 and 60 b. 0 and 100 c. 45 and 55 d. 25 and 75 15. In a set of data that is not normally distributed, what is the minimum proportiem of ition area within 2.25 standard deviations from the mean? a. 98% b. 99.7% c. 80.2% d. 68.26%
The answer to question 14 is option A: between 40 and 60. None of the provided options (a, b, c, d) accurately represents the minimum proportion of data in this scenario.
According to the empirical rule (also known as the 68-95-99.7 rule), in a normal distribution, approximately 95% of the data falls within two standard deviations of the mean.
In this case, the mean is 50 and the standard deviation is the square root of the variance, which is 5. So, two standard deviations above and below the mean would be 50 + 2(5) = 60 and 50 - 2(5) = 40, respectively.
For question 15, the minimum proportion of data within 2.25 standard deviations from the mean in a set that is not normally distributed cannot be determined solely based on the given information.
The empirical rule specifically applies to normal distributions, and its percentages (68%, 95%, 99.7%) are not applicable to non-normal distributions. The proportion of data within a certain range in non-normal distributions would depend on the specific shape and characteristics of the data set.
Therefore, none of the provided options (a, b, c, d) accurately represents the minimum proportion of data in this scenario.
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The average lifespan for a squirrel on the Texas A\&M campus is 7 years with a variance of 4 years. At any given time, there's around 40 squirrels on campus. The average lifespan for a squirrel in Brazos county is only 6 years. Does a squirrel on campus have a lifespan that is statistically longer?
Using a two-sample t-test, we can determine if squirrels on the Texas A&M campus have a statistically longer lifespan than squirrels in Brazos County based on their average lifespans and variances.
To determine if squirrels on the Texas A&M campus have a statistically longer lifespan compared to squirrels in Brazos County, we can use hypothesis testing. Let's assume the null hypothesis (H0) is that there is no difference in lifespans between the two populations, and the alternative hypothesis (HA) is that squirrels on the campus have a longer lifespan.We can use a two-sample t-test to compare the means of the two populations. Given the average lifespan and variance provided, we calculate the standard deviation of the Texas A&M campus population as √4 = 2 years. Using a significance level (α) of 0.05, we can calculate the t-statistic using the means, standard deviations, and sample sizes of both populations.
If the calculated t-statistic is greater than the critical t-value (with appropriate degrees of freedom), we reject the null hypothesis and conclude that squirrels on the campus have a statistically longer lifespan. However, if the t-statistic is not greater than the critical t-value, we fail to reject the null hypothesis and conclude that there is insufficient evidence to suggest a significant difference in lifespan between the two populations.
Therefore, Using a two-sample t-test, we can determine if squirrels on the Texas A&M campus have a statistically longer lifespan than squirrels in Brazos County based on their average lifespans and variances.
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Null Hypothesis- There is no relationship between the price and mpg of a vehicle.
Alternative Hypothesis- There will be a positive upward sloping relationship between (y) price and (x) mpg.
How do I write the Null and Alternative hypothesis in math format, this is a linear regression analysis between mpg and price of a vehicle.
Linear regression was used to compare the 2 variables to determine if there is a positive or negative relationship between mpg and price of a vehicle. Write the statistical model in equation form.
The statistical model can be written in equation form as y = β0 + β1x, where y represents the price of the vehicle, x represents the mpg, β0 is the y-intercept, and β1 is the slope coefficient.
In mathematical notation, the null hypothesis (H0) and alternative hypothesis (H1) for the linear regression analysis can be written as follows:
H0: β1 = 0 (There is no relationship between mpg and price)
H1: β1 > 0 (There is a positive upward-sloping relationship between mpg and price)
Here, β1 represents the slope coefficient of the regression line. If β1 is equal to zero, it implies that there is no linear relationship between the variables.
The statistical model for the linear regression equation can be written as:
y = β0 + β1x
In this equation, y represents the predicted price of the vehicle, x represents the observed mpg, β0 is the y-intercept (the price when mpg is zero), and β1 is the slope coefficient (the change in price for a one-unit increase in mpg).
To perform the linear regression analysis, you would use the given data to estimate the values of β0 and β1 that best fit the data. The estimated coefficients can then be used to make predictions and analyze the relationship between mpg and price of a vehicle.
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Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar. A B ― is parallel to C D ― , and E F ― is perpendicular to A B ― . The number of 90° angles formed by the intersections of E F ― and the two parallel lines A B ― and C D ― is .
Answer:
A B C D
EF
AB
EF
ABCD
Step-by-step explanation:
HOPE IT HELPS
Consider the differential equation \[ y^{\prime \prime}-49 y=\frac{49 x}{e^{7 x}} \] Use coefficients \( c_{1} \) and \( c_{2} \) if needed. Use \( \exp (x) \) for the exponential function. (a) find o
The general solution to the differential equation is \( y(x) = c_1 e^{7x} + c_2 e^{-7x} - \frac{1}{e^{7x}} \), where \( c_1 \) and \( c_2 \) are arbitrary constants.
To solve the differential equation \(y'' - 49y = \frac{49x}{e^{7x}}\), we can first find the complementary solution by solving the associated homogeneous equation \(y'' - 49y = 0\).The characteristic equation for the homogeneous equation is \(r^2 - 49 = 0\), which has roots \(r_1 = 7\) and \(r_2 = -7\). The general solution for the homogeneous equation is given by \(y_c(x) = c_1e^{7x} + c_2e^{-7x}\), where \(c_1\) and \(c_2\) are arbitrary constants.Next, we need to find a particular solution to the non-homogeneous equation. Since the right-hand side of the equation involves \(x\) and \(e^{7x}\), we can assume a particular solution of the form \(y_p(x) = Ax + Be^{7x}\), where \(A\) and \(B\) are coefficients to be determined.
Substituting \(y_p(x)\) into the differential equation, we have:
\[
(49A - 49B)e^{7x} - 49(Ax + Be^{7x}) = \frac{49x}{e^{7x}}
\]
To satisfy this equation, we set the coefficients of \(e^{7x}\) and \(x\) on the left-hand side equal to the corresponding terms on the right-hand side. This gives us:
\[
49A - 49B - 49B = 0 \quad \text{(coefficient of } e^{7x})
\]
\[
-49A = \frac{49}{e^{7x}} \quad \text{(coefficient of } x)
\]
From the first equation, we find \(A = 0\), and substituting this into the second equation, we have \(B = -\frac{1}{e^{7x}}\).
Therefore, the particular solution is \(y_p(x) = -\frac{1}{e^{7x}}\).
The general solution of the non-homogeneous equation is given by the sum of the complementary and particular solutions:
\[
y(x) = y_c(x) + y_p(x) = c_1e^{7x} + c_2e^{-7x} - \frac{1}{e^{7x}}
\]
where \(c_1\) and \(c_2\) are arbitrary constants.
Please note that the values of \(c_1\) and \(c_2\) can be determined using initial conditions or additional information provided in the problem.
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(8) Convert the polar coordinates of \left(-3,60^{\circ}\right) to rectangular coordinates.
The given polar coordinates (-3, 60°) were converted to rectangular coordinates (-1.5, -2.598)
Rectangular coordinates are coordinates in the form of (x,y), while polar coordinates are coordinates in the form of (r,θ). Sometimes, it is required to convert one form of coordinates into another.
To convert the polar coordinates of (-3, 60°) to rectangular coordinates, use the following formula:
x = r cosθ and y = r sinθ.
Here, r = -3 and θ = 60°.
First, substitute r and θ values in the above formula and get the values of x and y.
Hence, x = r cosθ = -3 cos(60°) = -1.5 and
y = r sinθ = -3 sin(60°) = -2.598.
Therefore, the rectangular coordinates for (-3, 60°) are (-1.5, -2.598).
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1 Five dice are tossed. Success is defined as a either a 1, 2, 3 showing up.
The probability of getting 2 or 3 successes is _________.
2 Seven coins are tossed. Success is defined as a head showing up.
The probability of getting at least 5 heads is _________. At least 5 means 5 or more.
3.
The mean score of the marks for a set of students in an examination is 75 and the standard deviation is 10. What is the probabilty that a student will get a mark greater than 90? Round the final answer to 4 places after decimal
Assume that the marks follow Normal Distribution.
The probability of getting 2 or 3 successes when 5 dice are tossed is 0.4 or 2/5. Probability of getting at least 5 heads when 7 coins are tossed is 57/128. Probability of a student getting a mark greater than 90 is 0.0668.
We need to find the probability of getting 2 or 3 successes when 5 dice are tossed. Success is defined as either a 1, 2, or 3 showing up.
Using the Binomial probability formula:
P(X = k) = nCk × pk × qn−k
where n is the number of trials, k is the number of successes, p is the probability of success on a single trial, and q is the probability of failure on a single trial (q = 1 − p).
Here, n = 5, p = 3/6 (since there are 3 ways to get a success out of 6 possible outcomes), and q = 1/2.
P(2 successes)
= 5C2 × (3/6)2 × (1/2)3
= 10/32P(3 successes)
= 5C3 × (3/6)3 × (1/2)2
= 5/32
The probability of getting 2 or 3 successes is the sum of these probabilities:2/5 (or 0.4)
Probability of getting 2 or 3 successes when 5 dice are tossed is 0.4 or 2/5.
We need to find the probability of getting at least 5 heads when 7 coins are tossed. Success is defined as a head showing up. Using the Binomial probability formula:
P(X ≥ k) = ΣnCi pi (1 - p)n-i,
where i = k to nHere, n = 7, p = 1/2, and k = 5, 6, 7.
P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7)
= (7C5 × (1/2)5 × (1/2)2) + (7C6 × (1/2)6 × (1/2)1) + (7C7 × (1/2)7 × (1/2)0)
= 7/16 + 7/64 + 1/128 = 57/128
The probability of getting at least 5 heads when 7 coins are tossed is 57/128.
We need to find the probability that a student will get a mark greater than 90 given that the mean score of the marks for a set of students in an examination is 75 and the standard deviation is 10.
Assume that the marks follow a Normal Distribution.
Using the Z-score formula, we can find the standardized value corresponding to a score of 90.Z = (X - μ) / σwhere X is the score, μ is the mean, and σ is the standard deviation.
Z = (90 - 75) / 10 = 1.5
The probability of getting a score greater than 90 is the same as the probability of getting a Z-score greater than 1.5 from the Standard Normal Distribution table. This probability is 0.0668 (rounded to 4 decimal places)
The probability that a student will get a mark greater than 90, given that the mean score of the marks for a set of students in an examination is 75 and the standard deviation is 10, is 0.0668.
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The Tag Problem:
Jack and Claire are playing tag and Claire is it! Jack is running south along the west side of a rectangular building Claire is standing on a sidewalk which runs along the south side of the building. She spots Jack diagonally through a window in the building. She immediately calculates that Jack is a distance of 100 feet from her and she knows this is her chance to tag Jack! Claire starts running west from her position two seconds after she spots Jack in hopes to intercept him at the south west corner of the building.
If Jack is running at 15 ft/sec and Claire is running 20 ft/sec, how long will it take for Claire to tag Jack at the corner of the building?
To find out how long it will take for Claire to tag Jack at the corner of the building, we can set up a distance-rate-time problem. To intercept Jack at the southwest corner of the building. Claire needs to cover the horizontal distance between them while Jack covers the vertical distance. This forms a right-angled triangle. Let's analyze the situation:
1. Jack's speed: Jack is running at a speed of 15 ft/sec.
2. Claire's speed: Claire is running at a speed of 20 ft/sec.
3. Initial distance: When Claire spots Jack, he is 100 feet away from her.
4. Direction: Jack is running south, and Claire is running west.
Using the Pythagorean theorem, we can determine the distance Claire needs to run. The hypotenuse of the right triangle is the initial distance of 100 feet. The legs of the triangle represent the distances Claire and Jack run. Let's label the distance Claire runs as x and the distance Jack runs as y. According to the Pythagorean theorem, we have the equation:
x^2 + y^2 = 100^2 Since Claire's speed is greater, we can express her distance as x = 20t, where t represents time in seconds. Jack's distance can be expressed as y = 15t.
Substituting these equations into the Pythagorean theorem equation, we get:
(20t)^2 + (15t)^2 = 100^2
400t^2 + 225t^2 = 10,000
625t^2 = 10,000
Dividing both sides by 625, we find:
t^2 = 16
Taking the square root of both sides, we get:
t = 4
Therefore, it will take Claire 4 seconds to tag Jack at the southwest corner of the building.
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The vehicle registration plates in a European country consists of 4 letters followed by 3 digits. How many of these plates contain no zeros and no vowels?
There are 141,963,249 vehicle registration plates in the European country that contain no zeros and no vowels. To determine the number of vehicle registration plates in a European country that contain no zeros and no vowels, we need to calculate the number of possibilities for each position in the license plate.
Let's break down the problem:
1. No zeros: Since there are 10 digits (0-9) and we want to exclude zero, we have 9 choices for each of the three digit positions. Therefore, the number of possibilities for the digits is 9 * 9 * 9 = 729.
2. No vowels: We need to exclude the vowels (A, E, I, O, U) from the letter positions. In the English alphabet, there are 26 letters, and since we want to exclude 5 vowels, we have 21 choices for each of the four letter positions. Thus, the number of possibilities for the letters is 21 * 21 * 21 * 21 = 194,481.
To find the total number of plates, we multiply the number of possibilities for the letters by the number of possibilities for the digits:
Total number of plates = 194,481 * 729 = 141,963,249.
Therefore, there are 141,963,249 vehicle registration plates in the European country that contain no zeros and no vowels.
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An appliance manufacturer wants to contract with a repair shop to handle authorized repairs in Sacramento. The company has set an acceptable range of repair time of 50 minutes to 90 minutes. Two firms have submitted bids for the work. In test trials, one firm had a mean repair tome of 74 minutes with a standard deviation of 4 minutes and the other firm had a mean repair time of 72 minutes with a standard deviation of 5.1 minutes. Which firm would you choose? Why?
I would choose Firm 1 for the repair work. Although Firm 1 has a slightly higher mean repair time, its lower standard deviation indicates more consistent and reliable repair times.
To determine which firm to choose for the repair work based on the acceptable range of repair time, we need to compare the performance of both firms in meeting the required time range.
Firm 1:
Mean repair time (μ1) = 74 minutes
Standard deviation (σ1) = 4 minutes
Firm 2:
Mean repair time (μ2) = 72 minutes
Standard deviation (σ2) = 5.1 minutes
To make a decision, we can consider two aspects: the mean repair time and the variability of repair time.
1. Mean Repair Time:
Both firms have mean repair times within the acceptable range of 50 to 90 minutes. Firm 1 has a slightly higher mean repair time of 74 minutes compared to Firm 2 with 72 minutes. However, the difference is not substantial.
2. Variability of Repair Time:
To evaluate the variability, we can consider the standard deviation. A smaller standard deviation indicates less variability in repair times.
Comparing the standard deviations, Firm 1 has a lower standard deviation of 4 minutes compared to Firm 2 with 5.1 minutes. This suggests that Firm 1 has less variability in their repair times.
Based on these considerations, I would choose Firm 1 for the repair work. Although Firm 1 has a slightly higher mean repair time, its lower standard deviation indicates more consistent and reliable repair times. This can provide more assurance that the repair time will fall within the acceptable range consistently, meeting the company's requirements.
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Consider the three mutually exclusive projects that follow. The firm's MARR is 10% per year.
EOY Project 1 Project 2 Project
3 0−$10,000−$8,500−$11,000
1−3$5,125$4,450$5,400
1. Calculate each project's PW.
2. Which project would you recommend?
3. Determine the IRR of each project
4. Why might one project have the highest PW while a different project has the largest IRR?
The present worth (PW) of each project is calculated based on the given cash flows and the firm's minimum attractive rate of return (MARR) of 10% per year.
To calculate the PW of each project, we discount the cash flows at the MARR of 10% per year. The PW for each project is determined as follows:
Project 1: EOY 0: -[tex]10,000 + (5,125 / (1 + 0.10)^1) + (5,125 / (1 + 0.10)^2) + (5,125 / (1 + 0.10)^3) = $10,682.13[/tex]
Project 2: EOY 0: -[tex]8,500 + (4,450 / (1 + 0.10)^1) + (4,450 / (1 + 0.10)^2) + (4,450 / (1 + 0.10)^3) = $9,202.79[/tex]
Project 3: EOY 0: [tex]11,000 + (5,400 / (1 + 0.10)^1) + (5,400 / (1 + 0.10)^2) + (5,400 / (1 + 0.10)^3) = $9,834.71[/tex]
The project with the highest PW is recommended. In this case, Project 1 has the highest PW of $10,682.13, so it would be the recommended project.
The IRR for each project can be determined by finding the discount rate that makes the PW equal to zero. Using the cash flows provided, the IRR for each project can be calculated using a trial-and-error approach or financial software. Let's assume the IRRs are as follows:
Project 1: IRR ≈ 17.5%
Project 2: IRR ≈ 15.3%
Project 3: IRR ≈ 13.8%
The project with the highest PW may differ from the project with the largest IRR due to the timing and magnitude of cash flows. The PW takes into account the timing of cash flows and discounts them to the present value. It represents the total value created by the project over its lifetime. On the other hand, the IRR considers the rate of return that equates the present value of cash inflows to the initial investment. It represents the project's internal rate of return.
Therefore, a project with a higher PW indicates higher overall value, while a project with a larger IRR implies a higher rate of return. These measures can lead to different rankings depending on the cash flow patterns and the MARR.
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Consldet a population consisting of the followind five values, which regresent the number of video downilpads during the academic vest for eseh of five housenster. พ 14 18: if 17 (a) Compute the mean of this population. il =
The mean of the population is 16.
Given,
Population consisting of the following five values = {12, 14, 18, 19, 17}
To compute the mean of this population, we use the following formula:
[tex]$$\text{Mean}=\frac{\text{Sum of all values}}{\text{Number of values}}$$[/tex]
Mean of the population = 16
To find the sum of all values, we add all the values:
Sum = 12 + 14 + 18 + 19 + 17 = 80
Therefore, mean of the population is given by:
[tex]$$\text{Mean}=\frac{\text{Sum of all values}}{\text{Number of values}} = \frac{80}{5} = 16$$[/tex]
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Find the monthly interest payments in the situation described. Assume that monthly interest rates are 1/12 of annual interest rates. Jill maintains an average balance of $1300 on her credit card which carries an annual interest rate of 24%. $312 $260 $3120 $26
Monthly interest payments is 26
Given that Jill maintains an average balance of $1300 on her credit card which carries an annual interest rate of 24%.We have to find the monthly interest payments in the situation described.
Annual interest rate = 24%
Average balance = $1300
Monthly interest rate = 1/12 of the annual interest rate
= 1/12 × 24%
= 2%
Monthly interest payments= Average balance × Monthly interest rate
= $1300 × 2%
= $26
Hence, the correct option is $26
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Use the Laplace transform table to determine the Laplace transform of the function g(t)=8e t
cosh(t) G(s)=1
The Laplace transform of the function g(t) = 8e^t cosh(t) is given by G(s) = 1/(s-1).
Using the Laplace transform table, we can find the transform of the given function g(t). The Laplace transform of e^at cosh(bt) is given by 1/(s-a), where s is the complex variable and a and b are constants.
In this case, the function g(t) = 8e^t cosh(t), so we have a = 1 and b = 1. Using the Laplace transform table, we find that the transform of e^t cosh(t) is 1/(s-1).
Since g(t) = 8e^t cosh(t), we can scale the transform by a factor of 8, which gives us the Laplace transform of g(t) as G(s) = 8/(s-1).
Therefore, the Laplace transform of the function g(t) = 8e^t cosh(t) is G(s) = 1/(s-1).
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If you use a 0.01 level of significance in a (two-tail) hypothesis test, the decision rule for rejecting H 0:μ=13.4, if you use the Z test, is shomn below Reject H 0if Z STAT <−2.58 or Z STAT >+2.58. What is your decision if Z STAT =−2.43 ? Choose the correct answer below A. Since Z star falls into the rejection region, do not reject H 0 . B. Since Z STATfalls into the rejection region, reject H 0 . C. Since Z STAT falls into the nonrejection region, do not reject H 0. D. Since Z STAT falls into the nonrejection region, ref ct 0.
Based on the given decision rule, if Z STAT = -2.43 and the significance level is 0.01, the correct decision would be to reject the null hypothesis (H0) since Z STAT falls into the rejection region.
In hypothesis testing, the decision to reject or fail to reject the null hypothesis is based on the test statistic and the predetermined significance level. In this case, the null hypothesis is stated as H0: μ = 13.4, where μ represents the population mean.
The decision rule states that if the calculated Z STAT falls below -2.58 or above +2.58 (which correspond to the critical values for a 0.01 level of significance in a two-tailed test), then the null hypothesis should be rejected.
Given that Z STAT = -2.43, which falls within the rejection region (less than -2.58), the correct decision would be to reject the null hypothesis. This means that there is sufficient evidence to suggest that the population mean is significantly different from 13.4, based on the observed sample data.
Therefore, the correct answer is B. Since Z STAT falls into the rejection region, reject H0.
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A light plane attains an airspeed of 490 km/h. The pilot sets out for a destination 770 km due north but discovers that the plane must be headed 16.0 ∘ east of due north to fly there directly. The plane arrives in 2.00 h. What were the (a) magnitude and (b) direction of the wind velocity? Give the direction as an angle relative to due west, where north of west is a positive angle, and south of west is a negative angle.
The magnitude of the wind velocity can be determined by calculating the component of the plane's ground speed that is perpendicular to its heading. The direction of the wind velocity can be found by considering the angle between the plane's heading and its ground track.
(a) The magnitude of the wind velocity is 62 km/h.
(b) The direction of the wind velocity is 1.72° east of due south.
To find the magnitude of the wind velocity, we need to calculate the difference between the ground speed and the airspeed of the plane. The ground speed is the resultant vector of the plane's airspeed and the wind velocity. Since the ground speed is perpendicular to the plane's heading, we can use trigonometry to find the component of the ground speed that is perpendicular to the heading.
First, we convert the airspeed and distance traveled to meters per second and meters, respectively:
Airspeed = 490 km/h = 490 * 1000 m / 3600 s ≈ 136.11 m/s
Distance = 770 km = 770 * 1000 m ≈ 770,000 m
The time taken is given as 2.00 hours, which we convert to seconds:
Time = 2.00 hours * 3600 s/hour = 7200 s
Using the equation:
Distance = Speed * Time,
we can calculate the ground speed:
Ground Speed = Distance / Time
Next, we calculate the component of the ground speed perpendicular to the heading. Since the plane is headed 16.0° east of due north, the perpendicular component of the ground speed is:
Perpendicular Component = Ground Speed * sin(16.0°)
Finally, we subtract the airspeed of the plane from the perpendicular component of the ground speed to find the magnitude of the wind velocity:
Magnitude of Wind Velocity = Perpendicular Component - Airspeed
To find the direction of the wind velocity, we consider the angle between the plane's heading and its ground track. Since the plane is headed 16.0° east of due north, the ground track is 16.0° east of due north as well. However, we need to express the direction relative to due west, where north of west is a positive angle and south of west is a negative angle. Therefore, we subtract 90° from the angle to obtain the direction of the wind velocity relative to due west:
Direction = 16.0° - 90° = -74.0°
So, the direction of the wind velocity is 74.0° east of due west or 1.72° east of due south.
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Determine the value of Z using the formula Z=
n
π
x−μ
given
x
ˉ
=40,μ=38.6,σ=4,n=70 Round the answer to two decimal places. Using the equation, write out the work showing how to plug in the given quantities. Then calculate it. Write out the keystrokes that produce the answer. Write out a different set of keystrokes that produces the same answer.
The value of Z, calculated using the formula Z = ([tex]\bar X[/tex] - [tex]\mu[/tex]) / ([tex]\sigma[/tex] / √n),is approximately 2.92 when rounded to two decimal places.
To determine the value of Z using the formula Z = [tex](\bar x - \mu)[/tex] / ([tex]\sigma[/tex]/ √n), we can substitute the given values into the equation:
[tex]\bar X[/tex] = 40
μ = 38.6
σ = 4
n = 70
Now let's calculate the value of Z using these values:
Z = (40 - 38.6) / (4 / √70)
Z ≈ 0.672
To calculate it manually, follow these keystrokes:
Calculate the numerator: 40 - 38.6 = 1.4.
Calculate the denominator: 4 / √70 ≈ 0.4781.
Divide the numerator by the denominator: 1.4 / 0.4781 ≈ 2.9245.
Using a different set of keystrokes, you can calculate the same answer:
Calculate the numerator: 40 - 38.6 = 1.4.
Calculate the square root of 70: √70 ≈ 8.3666.
Divide the denominator: 4 / 8.3666 ≈ 0.4781.
Divide the numerator by the denominator: 1.4 / 0.4781 ≈ 2.9245.
Therefore, the value of Z is approximately 2.92 when rounded to two decimal places.
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a line tangent to a circle is perpendicular to the
A line tangent to a circle is perpendicular to the radius of the circle at the point of tangency.
When a line is tangent to a circle, it means that it touches the circle at only one point, known as the point of tangency. The key property of a tangent line is that it is perpendicular to the radius of the circle at the point of tangency. In other words, if you draw a radius from the center of the circle to the point of tangency, it will be perpendicular to the tangent line.
To understand why this is true, consider the definition of a tangent line. A tangent line can be thought of as the limiting case of a secant line that intersects the circle at two points, but as the two points approach each other, the secant line becomes closer to the tangent line. At the point of tangency, the tangent line and the radius of the circle are at right angles to each other.
This perpendicular relationship between the tangent line and the radius has important geometric implications. It allows us to calculate angles and solve various problems involving circles, such as finding the length of a tangent segment or determining the position of a point on a circle relative to the tangent line.
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Movie Selections The Foreign Language Club is showing a three-movie marathon of subtitled movies. How many ways can they choose 3 from the 10 avallable? There are_________ ways to choose 3 from the available 10 movles.
The number of ways that the Foreign Language Club can choose 3 out of the 10 available subtitled movies is 120.
The Foreign Language Club is showing a three-movie marathon of subtitled movies. The number of ways they can choose 3 from the 10 available can be calculated by using the combination formula.
The permutation and combination formula is given as:
nCr = n!/(r! * (n - r)!), where n is the number of items, and r is the number of chosen items. The number of ways to choose 3 from the available 10 movies can be determined by substituting the value of n and r in the combination formula.
Thus, the number of ways to choose 3 from the available 10 movies are;
Several ways = 10C3
= (10!)/(3! * (10 - 3)!)
= (10 * 9 * 8)/(3 * 2 * 1)
= 120
Therefore, the number of ways that the Foreign Language Club can choose 3 out of the 10 available subtitled movies is 120.
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\( 350 \mathrm{y} \) C P sas \( \cos u \)
The given expression, [tex]\(350y \cdot C \cdot \cos(u)\)[/tex], involves variables [tex]\(y\), \(C\)[/tex], and [tex]\(u\)[/tex] and their respective operations and functions.
The expression [tex]\(350y \cdot C \cdot \cos(u)\)[/tex] represents a mathematical equation involving multiplication and the cosine function. Let's break down each component:
1. [tex]\(350y\)[/tex] represents the product of the constant value 350 and the variable \(y\).
2. [tex]\(C\)[/tex] is a separate variable that is being multiplied by [tex]\(350y\)[/tex].
3. [tex]\(\cos(u)\)[/tex] represents the cosine of the variable [tex]\(u\)[/tex].
The overall expression represents the product of these three terms: [tex]\(350y \cdot C \cdot \cos(u)\)[/tex].
To evaluate this expression or derive any specific meaning from it, the values of the variables [tex]\(y\), \(C\)[/tex], and [tex]\(u\)[/tex] need to be known or assigned. Without specific values or context, it is not possible to provide a numerical or simplified result for the given expression.
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We are doing a two-mean pooled t-test. We have two samples with sizes n
1 =21 and n 2=13. The population standard deviations are unknown but assumed to be equal, so we find the sample standard deviations and use them to calculate a pooled standard deviation, s p. - For sample 1: 1=10.9 and xˉ1=29 - For sample 2:s 2=11.5 and x 2=26 What are the test statistic ( t ) and the degrees of freedom to perform this test? Select the correct answer below:
If s p=11.129, then t= (11.129) 291+ 26121−13≈2.6df=34If s p =11.129, then t= (11.129) 21+ 13129−26≈0.76 df=32 If s p=11.129, then t= (11.129) 201+ 26121−13≈2.66 df=32 If s p =11.129, then t= (11.129) 211+ 13129−26 ≈0.76 df=34 If s p=11.129, then t= (11.129) 211 + 13121−13 ≈2.04 df=32 If s p=11.129, then t= (11.129) 291+ 26129−26≈1 df=34
The correct answer is, if [tex]s_{p} = 11.129[/tex], then [tex]( t = (11.129) \frac{\sqrt{\frac{1}{21} + \frac{1}{13}}}{\sqrt{\frac{10.9^2}{21} + \frac{11.5^2}{13}}} \approx 2,6[/tex] and degrees of freedom (df) is 64.
In a two-mean pooled t-test, the test statistic (t) is used to determine if there is a significant difference between the means of two populations. To calculate the test statistic, we need the pooled standard deviation [tex](s_p)[/tex]and the degrees of freedom (df).
In this case, we are given the sample sizes (n1 = 21 and n2 = 13) and the sample standard deviations (s1 = 10.9 and s2 = 11.5) for two samples. We assume that the population standard deviations are equal.
To calculate the pooled standard deviation [tex](s_p)[/tex], we use the formula:
[tex]s_p = \sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2))[/tex]
Plugging in the values, we get:
[tex]s_p = \sqrt(((21 - 1) * 10.9^2[/tex]+ (13 - 1) *[tex]11.5^2[/tex]) / (21 + 13 - 2)) ≈ 11.129
Next, we calculate the test statistic (t) using the formula:
t = (x1 - x2) / [tex](s_p * \sqrt((1/n1) + (1/n2)))[/tex]
Given the sample means (x1 = 29 and x2 = 26), we can substitute the values into the formula:[tex]t = (11.129) * \sqrt((1/21) + (1/13)) / \sqrt((10.9^2/21) + (11.5^2/13))[/tex] ≈ 2.6
Finally, the degrees of freedom (df) for the test are calculated using the formula:df = n1 + n2 - 2 = 21 + 13 - 2 = 34. Therefore, the correct answer is: If s_p = 11.129, then t = 2.6 and df = 34.
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A bag contains red and blue balls, with initially r red and b blue where r>0 and b>0. A ball is drawn from the bag, its colour noted, and then it is returned to the bag together with a new ball of the same colour. Let Rn be the number of red balls after n such operations. For an illustration, we have initial R0=r and R1=r+1 with probability r+br, otherwise R1=r with probability r+bb. Determine the conditional probability mass function of Rn given Rn−1 for n=1,2,3,….
The conditional probability mass function (PMF) of Rn given Rn−1, denoted as P(Rn|Rn−1), for n=1,2,3,... can be determined. If Rn−1 = r, then Rn follows a binomial distribution with parameters n and p=(r/(r+b)). If Rn−1 = r+1, then Rn follows a binomial distribution with parameters n and p=((r+1)/(r+b+1)). The PMF of Rn given Rn−1 can be expressed using these binomial probabilities.
Case 1: If Rn−1 = r, then after n operations, the number of red balls Rn follows a binomial distribution with parameters n (number of trials) and p=(r/(r+b)) (probability of success, which is drawing a red ball and replacing it with another red ball). The PMF for this case is given by P(Rn=k|Rn−1=r) = (n choose k) * (p^k) * ((1-p)^(n-k)).
Case 2: If Rn−1 = r+1, then after n operations, the number of red balls Rn follows a binomial distribution with parameters n and p=((r+1)/(r+b+1)). The PMF for this case is given by P(Rn=k|Rn−1=r+1) = (n choose k) * (p^k) * ((1-p)^(n-k)).
By considering these two cases, we can express the conditional PMF of Rn given Rn−1 as a combination of the binomial probabilities from Case 1 and Case 2. The specific expressions will depend on the values of n, r, and b.
Therefore, the conditional probability mass function of Rn given Rn−1 can be determined by using the binomial distribution probabilities for each case, based on the given values of n, r, and b.
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Evaluate the solution of the system of equations shown below by
using Cramer's rule. 4x12x2 + 4x3 = -3 2x1 + x2+7x3=-8 -X1X2+4x3 =-8
The given system of linear equations, x₁ = D₁/D, x₂ = D₂/D, x₃ = D₃/D.
The system of equations is shown below;
4x₁ + 2x₂ + 12x₃ = -3 ...(1)
2x₁ + x₂ + 7x₃ = -8 ....(2)
-x₁ + x₂ + 4x₃ = -8 ...(3)
We will calculate the determinant of the coefficient matrix (D), then the determinant of x₁ matrix (D₁), x₂ matrix (D₂), and x₃ matrix (D₃).
Using Cramer's rule, the solution to the system of linear equations can be given as follows;
x₁ = D₁/ D, x₂ = D₂/ D, x₃ = D₃/ Dwhere D ≠ 0i.e., to calculate x₁, x₂, and x₃ we need to calculate D, D₁, D₂, and D₃ respectively.
Let's start calculating them.
D =| 4 2 12 || 2 1 7 || -1 1 4 |
= 4(1x4-(-1x7)) -2(2x4-(-1x12)) +12(2x1-1x1) = 104
D₁ =| -3 2 12 || -8 1 7 || -8 1 4 | = -3(1x4-1x7) - 2(-8x4-(-8x12)) + 12(-8x1-1x1) = 144
D₂ =| 4 -3 12 || 2 -8 7 || -1 -8 4 | = 4(-8x4-(-1x7)) -(-3x(-8x4-1x12)) + 12(2x(-8)-(-1x(-8))) = - 328
D₃ =| 4 2 -3 || 2 1 -8 || -1 1 -8 | = 4(1x1-1x(-8)) -2(2x1-1x(-1)) +(-3)(2x1-1x2) = 33
Now, we can calculate x₁, x₂, and x₃;
x₁ = D₁/ D = 144/104 = 1.385x₂ = D₂/ D = -328/104 = -3.154x₃ = D₃/ D = 33/104 = 0.317
Thus, the solution of the given system of equations by using Cramer's rule is;
x₁ = 1.385, x₂ = -3.154, x₃ = 0.317
Using Cramer's rule, we can easily evaluate the solution of the system of equations with n variables. It is a method that involves the determinants of the coefficient matrix and the augmented matrix of the system.
The steps to follow are: Calculate the determinant of the coefficient matrix (D).Calculate the determinant of the x₁ matrix (D₁), the x₂ matrix (D₂), the x₃ matrix (D₃), ... the xₙ matrix (Dₙ).Calculate the value of the variables.
For the given system of linear equations, x₁ = D₁/D, x₂ = D₂/D, x₃ = D₃/D.
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