Let x(t) denote the amount of salt (in kg) in the tank at time t. The tank starts with 18 kg of salt, so x (0) = 18.
The solution is drained from the tank at a rate of 90 L/min, so that the amount of salt in the tank changes according to the differential equation
dx(t)/dt = - (x(t) kg)/(9000 L) × (90 L/min) = -1/100 x(t) kg/min
or, more succintly,
x' = -1/100 x
This equation is separable as
dx/x = -1/100 dt
Integrating both sides gives
∫ dx/x = -1/100 ∫ dt
ln|x| = -1/100 t + C
x = exp(-1/100 t + C )
x = C exp(-t/100)
(a) Using the initial condition x (0) = 18, we find
18 = C exp(0) ==> C = 18
so that
x(t) = 18 exp(-t/100)
(b) After 20 minutes, we have
x (20) = 18 exp(-20/100) = 18 exp(-1/5) ≈ 14.74
so the tank contains approximately 14.74 kg of salt after this time.
Susan has calculated that she needs $58000 her first year of retirement to maintain her standard of living. She expects to receive
$1000 per month from her employer defined-benefit pension and $1500 per month from Social Security. What is her annual
retirement income shortfall?
answer options:
28,000
40,000
58,000
150,000
Answer:
$28000
Step-by-step explanation:
12×1000 + 12×1500 = 12000 + 18000 = $30000
the shortfall is the difference between $58000 and $30000
58000 - 30000 = $28000
Relate what you know about simplifying expressions to what you know about
factoring. For example, before you can factor 12x + 20y + y, you need to simplify it.
Explain why.
Write the equation in slope-intercept form.
y+3 - 2(x-1)
Answer:
y = 2x - 5
Step-by-step explanation:
[tex]y+3=2(x-1)\\y+3=2x-2\\y+3-3=2x-2-3\\y=2x-5[/tex]
Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.438.43 and a standard deviation of 1.51.5. Using the empirical rule, what percentage of American women have shoe sizes that are between 6.936.93 and 9.939.93
Answer:
The right solution is "68%".
Step-by-step explanation:
The empirical rule is:
[tex]X\sim N(8.43, 1.5)[/tex]
According to the question,
= [tex]P(6.93< \mu < 9.93)[/tex]
= [tex]P(\frac{6.93-8.43}{1.5} < \frac{\mu -8.43}{1.5} < \frac{9.93-8.43}{1.5} )[/tex]
= [tex]P(z(1)-P(z(-1))[/tex]
= [tex]68[/tex] (%)
Thus the above is the right solution.
Estimating Mean SAT Math Score
Type numbers in the boxes.
aby Part 1: 5 points
The SAT is the most widely used college admission exam. (Most community
aby Part 2: 5 points
colleges do not require students to take this exam.) The mean SAT math score
varies by state and by year, so the value of u depends on the state and the year. 10 points
But let's assume that the shape and spread of the distribution of individual SAT math scores in each
state is the same each year. More specifically, assume that individual SAT math scores consistently
have a normal distribution with a standard deviation of 100. An educational researcher wants to
estimate the mean SAT math score (u) for his state this year. The researcher chooses a random
sample of 661 exams in his state. The sample mean for the test is 494.
Find the 99% confidence interval to estimate the mean SAT math score in this state for this year.
(Note: The critical z-value to use, zc, is: 2.576.)
Your answer should be rounded to 3 decimal places.
Answer:
(483.981 ; 504.019)
Step-by-step explanation:
Given :
σ = 100
Sample size, n = 661
xbar = 494
We use the Z distribution since we are working with the population standard deviation ;
C.I = xbar ± (Zcritical * σ/√n)
Zcritical at 99% = 2.576
C.I = 494 ± (2.576 * 100/√661)
C.I = 494 ± 10.019
Lower boundary = (494−10.019) = 483.981
Upper boundary = (494+10.019) = 504.019
C.I = (483.981 ; 504.019)
3x+2y <
11 2x-y<9
Does this graph match this equation?
Answer:
No.
Step-by-step explanation:
3x+2y<11 doesn't have a y-intercept of 6 and doesn't have a x-intercept of 4.
2x-y<9 is not the same direction as 3x+2y<11.
Solve the following system of equations by graphing.
y = -x-1
y = 14x - 4
A) (-4,3)
B) (-3,4)
C) (4,-3)
D) (3,4)
Answer:
(0.2, -1.2)
Step-by-step explanation:
When solving a system of equations by graphing, we first plot the two equations on a graph, then the point of intersection of the two graphs is the solution to the system of equations.
Therefore giving the equations y = - x - 1; and y = 14x - 4, we have to first plot the both linear equations using online geogebra graphing tool. The intersection of both linear graphs is the solution to the problem.
We can see that the point of intersection is A(0.2, -1.2)
[{66 +1} 2-6].7
I need help ASAP please due Monday pre-algebra show work
Ans; 7× [2-6 { 1+66}] —> 7× [2 - 6 { 67} ] —> 7× [2-402] —> 7×[- 400] —> = – 2800
I hope I helped you ^_^
Can someone please help me with this
Answer:
d
Step-by-step explanation:
h
square root of v-5=6
[tex] \sqrt{x - 5 = 6}[/tex]
Answer:
I think you mean this :
[tex] \sqrt{x - 5} = 6 \\ = > {( \sqrt{x - 5} })^{2} = {6}^{2} \\ = > x - 5 = 36 \\ = > x = 36 + 5 \\ = > x = 41[/tex]
Or,
Square root of x-5=6 is :
[tex] \sqrt{x - 5} \:=\:\sqrt{6} [/tex]
What is the slope of a line perpendicular to line A?
About 3% of the population has a particular genetic mutation. 200 people are randomly selected.
Find the mean for the number of people with the genetic mutation in such groups of 200
Answer:
6
Step-by-step explanation:
200. Move decimal twice to the left. 1% of 200 is 2. 2*3 is 6.
What is the image of (3, -12) after a dilation by a scale factor of į centered
at the origin?
Answer:
9 is. ................m.m..mk
help me solve this trig
Hello there!
Previously, we learnt that to solve the equation, we have to isolate the sin, cos, tan, etc first.
First Question
The first question has sin both sides. Notice that if we move sin(theta) to left. We get:-
[tex] \displaystyle \large{2 {sin}^{2} \theta - sin \theta = 0}[/tex]
We can common factor out the expression.
[tex] \displaystyle \large{sin \theta(2sin \theta - 1) = 0}[/tex]
It is a trigonometric equation in quadraric pattern.
We consider both equations:-
First Equation
[tex] \displaystyle \large{sin \theta = 0}[/tex]
Remind that sin = y. When sin theta = 0. It means that it lies on the positive x-axis.
We know that 0 satisfies the equation, because sin(0) is 0.
Same goes for π as well, but 2π does not count because the interval is from 0 ≤ theta < 2π.
Hence:-
[tex] \displaystyle \large { \theta = 0,\pi}[/tex]
Second Equation
[tex] \displaystyle \large{2sin \theta - 1 = 0}[/tex]
First, as we learnt. We isolate sin.
[tex] \displaystyle \large{sin \theta = \frac{1}{2} }[/tex]
We know that, sin is positive in Quadrant 1 and 2.
As we learnt from previous question, we use π - (ref. angle) to find Q2 angle.
We know that sin(π/6) is 1/2. Hence π/6 is our reference angle. Since π/6 is in Q1, we only have to find Q2.
Find Quadrant 2
[tex] \displaystyle \large{\pi - \frac{\pi}{6} = \frac{6\pi}{6} - \frac{\pi}{6} } \\ \displaystyle \large{ \frac{5\pi}{6} }[/tex]
Hence:-
[tex] \displaystyle \large{ \theta = \frac{\pi}{6} , \frac{5\pi}{6} }[/tex]
Since both first and second equations are apart of same equation. Therefore, mix both theta from first and second.
Therefore, the solutions to the first question:-
[tex] \displaystyle \large \boxed{ \theta = 0,\pi, \frac{\pi}{6} , \frac{5\pi}{6} }[/tex]
Second Question
This one is a reciprocal of tan, also known as cot.
[tex] \displaystyle \large{cot3 \theta = 1}[/tex]
For this, I will turn cot to 1/tan.
[tex] \displaystyle \large{ \frac{1}{tan3 \theta} = 1}[/tex]
Multiply whole equation by tan3 theta, to get rid of the denominator.
[tex] \displaystyle \large{ \frac{1}{tan3 \theta} \times tan3 \theta = 1 \times tan3 \theta } \\ \displaystyle \large{ 1= tan3 \theta }[/tex]
We also learnt about how to deal with number beside theta.
We increase the interval, by multiplying with the number.
Since our interval is:-
[tex] \displaystyle \large{0 \leqslant \theta < 2\pi}[/tex]
Multiply the whole interval by 3.
[tex] \displaystyle \large{0 \times 3 \leqslant \theta \times 3 < 2\pi \times 3} \\ \displaystyle \large{0 \leqslant 3 \theta < 6\pi }[/tex]
We also know that tan is positive in Quadrant 1 and Quadrant 3.
and tan(π/4) is 1. Therefore, π/4 is our reference angle and our first theta value.
When we want to find Quadrant 3, we use π + (ref. angle).
Find Q3
[tex] \displaystyle \large{\pi + \frac{\pi}{4} } = \frac{5\pi}{4} [/tex]
Hence, our theta values are π/4 and 5π/4. But that is for [0,2π) interval. We want to find theta values over [0,6π) interval.
As we learnt previously, that we use theta + 2πk to find values that are in interval greater than 2π.
As for tangent, we use:-
[tex] \displaystyle \large{ \theta + \pi k = \theta}[/tex]
Because tan is basically a slope or line proportional graph. So it gives the same value every π period.
Now imagine a unit circle, and make sure to have some basic geometry knowledge. Know that when values addition by 180° or π would give a straight angle.
We aren't using k = 1 for this because we've already found Q3 angle.
Since we know Q1 and Q3 angle in [0,2π).
We can also use theta + 2πk if you want.
First Value or π/4
[tex] \displaystyle \large{ \frac{\pi}{4} + 2\pi = \frac{9\pi}{4} } \\ \displaystyle \large{ \frac{\pi}{4} + 4\pi = \frac{17\pi}{4} }[/tex]
Second Value or 5π/4
[tex] \displaystyle \large{ \frac{5\pi}{4} + 2\pi = \frac{13\pi}{4} } \\ \displaystyle \large{ \frac{5\pi}{4} + 4\pi = \frac{21\pi}{4} }[/tex]
Yes, I use theta + 2πk for finding other values.
Therefore:-
[tex] \displaystyle \large{3 \theta = \frac{\pi}{4} , \frac{5\pi}{4} , \frac{9\pi}{4}, \frac{17\pi}{4} , \frac{13\pi}{4} , \frac{21\pi}{4} }[/tex]
Then we divide every values by 3.
[tex] \displaystyle \large \boxed{\theta = \frac{\pi}{12} , \frac{5\pi}{12} , \frac{9\pi}{12}, \frac{17\pi}{12} , \frac{13\pi}{12} , \frac{21\pi}{12} }[/tex]
Let me know if you have any questions!
Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out x apartments is given by the following function. p(x)=-12x^2+2160x-59000 To maximize the monthly rental profit, how many units should be rented out? units What is the maximum monthly profit realizable?
Answer:
To maximize the monthly rental profit, 90 units should be rented out.
The maximum monthly profit realizable is $38,200.
Step-by-step explanation:
Vertex of a quadratic function:
Suppose we have a quadratic function in the following format:
[tex]f(x) = ax^{2} + bx + c[/tex]
It's vertex is the point [tex](x_{v}, y_{v})[/tex]
In which
[tex]x_{v} = -\frac{b}{2a}[/tex]
[tex]y_{v} = -\frac{\Delta}{4a}[/tex]
Where
[tex]\Delta = b^2-4ac[/tex]
If a<0, the vertex is a maximum point, that is, the maximum value happens at [tex]x_{v}[/tex], and it's value is [tex]y_{v}[/tex].
In this question:
Quadratic equation with [tex]a = -12, b = 2160, c = -59000[/tex]
To maximize the monthly rental profit, how many units should be rented out?
This is the x-value of the vertex, so:
[tex]x_{v} = -\frac{b}{2a} = -\frac{2160}{2(-12)} = \frac{2160}{24} = 90[/tex]
To maximize the monthly rental profit, 90 units should be rented out.
What is the maximum monthly profit realizable?
This is p(90). So
[tex]p(90) = -12(90)^2 + 2160(90) - 59000 = 38200[/tex]
The maximum monthly profit realizable is $38,200.
Solve 2x2 – 3x = 12 using the quadratic formula.
Quadratic Formula: (-b +/- sqrt(b^2 - 4ac)) / 2a
2x^2 - 3x = 12
2x^2 - 3x - 12 = 0
a = 2
b = -3
c = -12
(--3 +/- sqrt( (-3)^2 - 4(2)(-12) )) / 2(2)
3 +/- sqrt( 9 + 96 ) / 4
3 +/- sqrt(105) / 4
Answers: [tex]\frac{3 + \sqrt{105} }{4}[/tex], [tex]\frac{3 - \sqrt{105} }{4}[/tex]
Hope this helps!
PLZ HELP ASAP
A student poll on campus wanted to analyze the correlation of the Number of calories consumed per day to the weight of a student. in the form of a paragraph describe which visual display is most appropriate to represent the data. explain your reasons for choosing this type of visual display.
Answer:
Each kids weight in a chart
Step-by-step explanation:
I chose this because its the most organized way of doing that
The age of Paul is 1/3 that of Kennedy. In four years time the age of Paul will be the same as Kennedy present age. How old is Paul now?
Answer:
Paul is 2 and Kennedy is 6
Step-by-step explanation:
6 × 1/3 = 2
2 + 4 =6
A population of values has a normal distribution with μ= 180.1
and σ=100. You intend to draw a random sample of size n=94
What is the mean of the distribution of sample means?
What is the standard deviation of the distribution of sample means?
(Report answer accurate to 2 decimal places.)
A sample of size n taken from a normally distributed population with mean µ and standard deviation σ has a sample mean of µ and standard deviation of σ/√n.
So the sample mean would still be 180.1, while the sample standard deviation would be 100/√94 ≈ 10.31.
ZDAC = ZBAD.
What is the length of BD?
Round to one decimal place.
Answer:
BD = 4.1
Step-by-step explanation:
DA is an angle bisector which also divides the opposite side of the angle it bisects in a way that it is proportional to tye other two sides.
By implication, we would have the following:
AB/BD = AC/DC
AB = 5.3
AC = 5.5
DC = 4.3
BD = ?
Plug in the values
5.3/BD = 5.5/4.3
Cross multiply
BD*5.5 = 4.3*5.3
BD*5.5 = 22.79
Divide both sides by 5.5
BD = 22.79/5.5
BD = 4.1 (to 1 decimal place)
How much does college tuition cost? That depends, of course, on where you go to college. Construct a weighted average. Using the data from "College Affordable for Most," estimate midpoints for the cost intervals. Say 46% of tuitions cost about $4,500; 21% cost about $7,500; 7% cost about $12,000; 8% cost about $18,000; 9% cost about $24,000; and 9% cost about $31,000. Compute the weighted average of college tuition charged at all colleges.
Answer:
0.127
Step-by-step explanation:
If f(x) = 5x squared -3 and g(x) = x squared - 4x -8, find (f-g)(x)
Answer:
[tex]4x^2+4x+5[/tex]
Step-by-step explanation:
[tex]f(x)=5x^2-3\\g(x)=x^2-4x-8[/tex]
Set up an expression.
[tex]5x^2-3-(x^2-4x-8)[/tex]
Distribute the negative (-1)
[tex]5x^2-3-x^2+4x+8[/tex]
Solve / Simplify
[tex]4x^2+4x+5[/tex]
I'm late, but I hope this helps!
2) Consider the quadratic sequence 72, 100, 120, 132
2.1.1) Determine Tn the nth term of the quadratic.
9514 1404 393
Answer:
Tn = -4n² +40n +36
Step-by-step explanation:
A graphing calculator readily performs the quadratic regression, yielding the formula ...
Tn = -4n² +40n +36
__
The first and second differences of the given sequence terms are ...
28, 20, 12 and -8, -8
The coefficient of the squared term is half the second difference, so is -4. Then the sequence of squared terms is -4n²:
-4, -16, -36, -64
Subtracting these values from the original sequence gives the linear sequence ...
76, 116, 156, 196
which has first term 76 and common difference 40. The equation for the n-th term of this is ...
an = 76 +40(n -1) = 36 +40n
Adding this linear sequence to the sequence of squared terms, we get ...
Tn = -4n² +40n +36
2xy+x+2y answer please
Step-by-step explanation:
2 x y + x + 2 y is equal to 3 x y + 2 y final answer is 5xy
A local grocery store receives strawberries from suppliers in Florida and California. Currently there are 18 strawberry containers on the shelf and 11 of them are from Florida. A shopper selects three containers to purchase. What is the probability that exactly one of the containers is from the Florida supplier
Using the hypergeometric distribution, it is found that there is a 0.2831 = 28.31% probability that exactly one of the containers is from the Florida supplier.
The containers are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.
Hypergeometric distribution:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
x is the number of successes. N is the size of the population. n is the size of the sample. k is the total number of desired outcomes.In this problem:
There are 18 containers, hence [tex]N = 18[/tex]11 of those are in Florida, hence [tex]k = 11[/tex].A sample of 3 containers is taken, hence [tex]n = 3[/tex]The probability that exactly one of the containers is from the Florida supplier is P(X = 1), hence:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]P(X = 1) = h(1,18,3,11) = \frac{C_{11,1}C_{7,2}}{C_{18,3}} = 0.2831[/tex]
0.2831 = 28.31% probability that exactly one of the containers is from the Florida supplier.
A similar problem is given at https://brainly.com/question/24826394
Find the integer pair that has the given product and sum. The product is 28 The sum is 11
Answer:
7 and 4
Step-by-step explanation:
when you multiply 7×4 the answer is 28 and when you add them the answer is 11.
I hope this helps
Answer:
4 and 7
Step-by-step explanation:
xy = 28 ..........1
x + y = 11........2
x = 11 - y.........3
substitute 3 in 1:
(11 - y)*y = 28
-y² + 11y - 28 = 0
y² - 11y + 28 = 0
( y - 7)( y - 4) = 0
y = 7 or y = 4
subs in 2:
x = 4 or x = 7
pairs: (4 ; 7) and (7 ; 4)
Given f (x) = 4x - 3,g(2) = x3 + 2x
Find (f - g) (4)
Factor the trinomial. Write the answer in binomial fctor form.
2x^2– 13x + 11
9514 1404 393
Answer:
(2x -11)(x -1)
Step-by-step explanation:
The middle term can be rewritten as a sum using the coefficients of the first and last terms. Then the expression can be factored by pairs of terms.
2x^2 -13x +11 = 2x^2 -2x -11x +11 = 2x(x -1) -11(x -1)
= (2x -11)(x -1)
find b for (b-1)/4=(7b+2)/12
Answer:
-1.25
Step-by-step explanation:
you first have to cross multiply
12(b-1)=4(7b+2)
12b-12=28b+8
group the like terms
12b-28b=8+12
-16b/-16=20/-16
b= -1.25
I hope it helps
Step-by-step explanation:
Answer is in the picture..
hope it helps
Given a committee of 8 women and 11 men, how many different ways are there to pick a female president, a male treasurer, and a secretary of either gender if one of the men, Pete, says that he cannot be the treasurer? Assume that none can hold more than one office.
Answer: 1360 ways.
Step-by-step explanation:
Number of men = 11
Number of women = 8
Total members = 8 + 11 = 19
Firstly, the number of ways of selecting 1 woman for female president out of 8 will be:
= 8C1
= 8
Since Pete can not be a treasurer, then the treasurer will then be selected from the remaining 10(11 - 1). The number of ways of selecting 1 treasurer out of 10 will be:
= 10C1
= 10
Thirdly, since none can hold more then one office, in this case, the selection will be done by selecting 1 person out of 7 women and 10 men. Therefore, the number of ways of selecting 1 person out of 17 will be:
= 17C1
= 17
Therefore, the total number of ways will then be:
= 8 × 10 × 17
= 1360
