Evaluate the definite integral. (Give an exact answer. Do not round.) ∫3 0 e^2x dx

The water level is rising at a rate of 0.006 feet per minute. This can be answered by the concept of Differentiation.

The formula for the volume of a rectangular box is V = lwh, where l, w, and h represent the length, width, and height respectively. Since the base of the bathtub is 28 square feet, we can assume that the length and width are both 28 feet. Let's say the height of the water in the bathtub is h at time t.

We know that the water is filling the bathtub at a rate of 5 cubic feet per minute, so the rate of change of the volume of water in the bathtub is 5. We want to find the rate of change of the height of the water, which we can call dh/dt.

Using the formula for the volume of a rectangular box, we can write:

V = lwh = 28wh

We can differentiate both sides with respect to time t:

dV/dt = 28w dh/dt

We know that dV/dt is 5, and w is also 28 since the base of the bathtub is a rectangle with sides of length 28 feet. Therefore, we can solve for dh/dt:

5 = 28(28) dh/dt

dh/dt = 5/(28×28)

dh/dt = 0.006 ft/min

Therefore, the water level is rising at a rate of 0.006 feet per minute.

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The graph of the function f(x) = 32 is attached accordingly.

X  - Intercept - There is no x-intercept since the function is a horizontal line.

Y -Interept - The y-intercept is (0, 32), since the line intersects the y-axis at y = 32.

  Vertical Asymptotes - There are 0    vertical asymptotes,   since t function is defined for all values of x.

Horizontal Asymptotes - There are 0    horizontal   asymptotes, since the function is a horizontal line.

Slant Asymptotes - There are zeroslant asymptotes, since the function is a horizontal line.

Holes - There are zeroholes in the graph, since the function is a horizontal line with no breaks or discontinuities.

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Answer: see below

Step-by-step explanation:

got it right on quiz

The amount that Julian will save, when Julian makes $500 is $95

The scattered points on the graph represents the given parameter

When the line of the best fit is drawn on the scattered graph, it passes through the points

(175, 30) and (25, 0)

A linear equation is represented as

y = mx + c

Using the given points, we have

25m + c = 0

175m = c = 30

Subtract the equations

So, we have

150m = 30

Divide

m = 0.2

Solving for c, we have

c = -25m

c = -25 * 0.2

c = -5

So, the equation is

y = 0.2x - 5

When Julian makes $500, we have

x = 500

This gives

y = 0.2 * 500 - 5

Evaluate

y = 95

Hence, the savings at $500 is $95

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Answer:

Step-by-step explanation:

5/6 + 2/3 = ?

5/6 + 4/6 =

9/6

semplify

3/2 or 1.5 or 1 1/2

Sure! So, given e z = xyz, we can use the product rule of differentiation to find the partial derivatives of z with respect to x and y ∂z/∂x = yz + xz(∂y/∂x) .

∂z/∂y = xz + yz(∂x/∂y), Since there are no other given values or constraints, we cannot simplify these further. I hope this helps! Let me know if you have any other questions, Given the equation z = xyz, to find the partial derivatives ∂z/∂x and ∂z/∂y, we can use the following:

For ∂z/∂x, we differentiate z with respect to x while treating y as a constant:
∂z/∂x = y*(1) + xyz*(0) = y

For ∂z/∂y, we differentiate z with respect to y while treating x as a constant:
∂z/∂y = x*(1) + xyz*(0) = x
So, ∂z/∂x = y and ∂z/∂y = x.

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Answer:

23 dollar in total

Step-by-step explanation:

since 11.50 for one if u buy 2 it would be 23dollar total

Evaluating the exponential equation we can see that after 25 months there will be  1,441 rabbits after 25 months.

We know that the population of rabbits is modeled by the exponential equation below:

P(n) = 23*1.18^n

Where n is the number of months.

Then the population after 25 months is what we get when we evaluate the exponential equation in n = 25, we will get:

P(25) = 23*1.18^25 = 1,441

There will be 1,441 rabbits after 25 months.

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The correct Z-scores for which 90% of the distribution's area lies between -z and Z are D) (-1.645, 1.645).

The Z-score is a measure of how many standard deviations a particular value is from the mean of a distribution. In a standard normal distribution, which has a mean of 0 and a standard deviation of 1, the Z-score represents the number of standard deviations a value is away from the mean.

To find the Z-scores for which 90% of the distribution's area lies between -z and Z, we need to find the Z-scores that correspond to the 5th and 95th percentiles of the standard normal distribution.

Since the distribution is symmetric, we can find the Z-scores for the lower and upper tails of the distribution and use them to determine the range between -z and Z.

Using a standard normal distribution table or a Z-table calculator, we can find that the Z-score corresponding to the 5th percentile (i.e., the value of -z) is approximately -1.645, and the Z-score corresponding to the 95th percentile (i.e., the value of Z) is also approximately 1.645.

Therefore, the correct Z-scores for which 90% of the distribution's area lies between -z and Z are (-1.645, 1.645).

Note: The other options given in the question (A) (-1.96, 1.96), (B) (-2.33, 2.33), and (C) (-0.99, 0.99) do not correspond to the Z-scores for which 90% of the distribution's area lies between -z and Z. Option (D) (-1.645, 1.645) is the correct answer.

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The probability that a girl will win two weeks in a row is 24%.

Probability tells how many times something will happen or be present.

The probability of a girl winning in a given week is 17/35 since there are 17 girls and 35 students total. Assuming each week's selection is independent of previous selections, the probability of a girl winning two weeks in a row is (17/35) x (17/35) = 289/1225.

Rounding this to the nearest percent gives a probability of 24%.

Therefore, the probability is 24%.

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It's reasonable to assume the sample distribution's shape would be similar to the population distribution's shape. However, without more information, we cannot confirm the exact shape of the distribution.

The center of the data distribution is represented by the mean. According to the national Census Bureau, the mean number of people in a household for the entire population is 4.66.

The variability of the population distribution is represented by the standard deviation. In this case, the standard deviation provided by the Census Bureau is 1.94.

So, the center of the data distribution is 4.66, and the variability of the population distribution is 1.94.

Since the Census Bureau has used a random sample of 225 homes, the sample mean (4.8) and standard deviation (2.1) could be used to estimate the population mean and standard deviation. However, these sample statistics are not necessarily equal to the population parameters.

As for the shape of the data distribution, it's difficult to predict without more information about the distribution itself. If the data is normally distributed, the shape would be bell-shaped. If the sample is representative of the population, it's reasonable to assume the sample distribution's shape would be similar to the population distribution's shape. However, without more information, we cannot confirm the exact shape of the distribution.

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Rate of change in the side length when the side lengths are 12 feet is approximately -0.0009259 ft/min.

Explanation:-

To find the rate of change in the side length of the cube when the side lengths are 12 feet and the volume decreases at a rate of 0.4 ft³/min, follow these steps:

Step 1: The formula for the volume of a cube.
Volume (V) = side length³, or V = s³

Step 2: Differentiate both sides with respect to time (t) to find the relationship between the rates of change.
dV/dt = 3s²(ds/dt)

Step 3: Plug in the given information: dV/dt = -0.4 ft³/min (since the volume is decreasing), and s = 12 feet.

-0.4 = 3(12²)(ds/dt)

Step 4: Solve for ds/dt, the rate of change in the side length.

-0.4 = 3(144)(ds/dt)
-0.4 = 432(ds/dt)

ds/dt = -0.4/432

Step 5: Simplify the expression.

ds/dt ≈ -0.0009259 ft/min

So, the rate of change in the side length when the side lengths are 12 feet is approximately -0.0009259 ft/min.

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Maclaurin series for cos(x) to compute [tex]\cos(3) \approx 0.99862$.[/tex]

What is Maclaurin series?

The Maclaurin series is a special case of the Taylor series, which is a power series expansion of a function about 0. The Maclaurin series is obtained by setting the center of the Taylor series to 0. It is named after the Scottish mathematician Colin Maclaurin.

The Maclaurin series of a function f(x) is given by:

[tex]f(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + ... + (f^{(n)}(0)/n!)x^n + ...[/tex]

where [tex]f^{(n)}(0)[/tex] denotes the nth derivative of f evaluated at 0.

Using the Maclaurin series for [tex]$\cos(x)$[/tex], we have:

[tex]\cos(x) &= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}(x)^{2n}[/tex]

Substituting [tex]$x=3$[/tex] into this series, we get:

[tex]\cos(3) &= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}(3)^{2n}[/tex]

[tex]&= 1 - \frac{3^2}{2!} + \frac{3^4}{4!} - \frac{3^6}{6!} + \frac{3^8}{8!} - \cdots[/tex]

[tex]&\approx 0.99862 \quad\text{(correct to five decimal places)}[/tex]

Therefore, [tex]\cos(3) \approx 0.99862$.[/tex]

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Hence, a and c are true .

When two numbers are split, the modulo operation in computing the remainder  of the division. A modulo n is the remainder of the Euclidean division of two positive numbers, a and n, where a is the dividend and n is the divisor.

A congruence relation is an equivalence relation that is symbol of addition, subtraction, and multiplication. Congruence modulo n is one one map  connection. The symbol for congruence modulo n is: The brackets indicate that (mod n) applies to both sides of the equation, not only the right-hand side  of the equation.

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Answer:

A) Tamara's work is correct.

Step-by-step explanation:

To determine whether a function is even, odd, or neither, we need to check whether f(-x) is equal to f(x) or -f(x).

In this case, Tamara correctly found the expression for f(-x) in step 1, and then in step 2, she checked whether f(-x) is equal to f(x) or -f(x).

Since f(-x) is equal to f(x), Tamara correctly concluded that f is an even function.

The circumference of the circle x² + y² = 1 is 2π.

To show that the circumference of the circle x² + y² = 1 is 2π, we can use the arc length formula. The formula for arc length (s) in a circle is given by:

s = r × θ

where r is the radius of the circle and θ is the central angle in radians.

For the circle x² + y² = 1, the radius (r) is equal to 1 (since the equation is already in the standard form). To find the circumference, we need to find the arc length for a complete circle. A complete circle has a central angle of 2π radians. Therefore, we can plug these values into the arc length formula:

Circumference = s = r × θ
Circumference = 1 × 2π
Circumference = 2π

Thus, the circumference of the circle x² + y² = 1 is 2π.

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Answer:

2

Step-by-step explanation:

The product of the three numbers under the given circumstances is 49,483.

Let's start by setting up the equations based on the given information:

x + y + z = 165 (equation 1)

7x = n (equation 2)

y - 9 = n (equation 3)

z + 9 = n (equation 4)

We want to find the product of x, y, and z, which is simply:

x * y * z

We can use equations 2, 3, and 4 to substitute n in terms of y and z:

7x = y - 9 (substituting equation 3)

7x = z + 9 (substituting equation 4)

Now we can substitute these expressions for y and z into equation 1 to get an equation in terms of x:

x + (7x + 9) + (7x - 9) = 165

15x = 165

x = 11

Substituting x = 11 into equations 2, 3, and 4, we get:

7(11) = n

n = 68

y = n + 9 = 68 + 9 = 77

z = n - 9 = 68 - 9 = 59

Now we can calculate the product of x, y, and z:

x * y * z = 11 * 77 * 59 = 49,483

Therefore, the product of the three numbers is 49,483.

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Let X be the number of students who will graduate in 4 years out of 1025 students. The final answer of  probabilities is  [tex]P(X = 697)[/tex][tex]=0.080[/tex];  [tex]P(X \leq 685) = 0.123[/tex][tex]P(X \geq 650) = 0.997[/tex][tex]P(665 \leq X \leq 715) =0.826[/tex]

Then X follows a binomial distribution for finding the probabilities with n = 1025 and p = 0.65.

(a) [tex]P(X = 697) = (1025 choose 697) * (0.65)^697 * (1-0.65)^(1025-697)[/tex]=[tex]0.080[/tex]

(b) [tex]P(X ≤ 685)[/tex]= [tex]Σ_(k=0)^685 (1025 choose k) * (0.65)^k * (1-0.65)^(1025-k)[/tex]= [tex]0.123[/tex]

(c) [tex]P(X ≥ 650)[/tex] = [tex]1 - P(X < 650) = 1 - Σ_(k=0)^649 (1025 choose k) * (0.65)^k * (1-0.65)^(1025-k)[/tex]= [tex]0.997[/tex]

(d) [tex]P(665 ≤ X ≤ 715)[/tex]= [tex]Σ_(k=665)^715 (1025 choose k) * (0.65)^k * (1-0.65)^(1025-k)[/tex]≈[tex]0.826[/tex]

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The water level y(t) = √(1000 - 80πt/3), te ≈ 11.8 seconds.

The water level y(t) in the cylindrical tank with radius 10 ft and length 40 ft decreases over time until the tank is empty at time t=te seconds can be found shown below:

First, find the volume of the half-filled tank: V = (1/2)π(10^2)(40) = 2000π ft³. The leakage rate Q = (1 in²)(1/144 ft²/in²) = 1/144 ft². Since Q = dV/dt, we have dV = -Qdy.

Integrating both sides gives V = -Qy + C. Initially, V = 2000π and y = 10, so C = 3000π. Thus, V = -Qy + 3000π. Solving for y, we get y(t) = √(1000 - 80πt/3). To find te, set V = 0 and solve for t: 0 = -80πt/3 + 1000, which gives te ≈ 11.8 seconds.

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1. Multiply both sides by the inverse of p on the left: p^(-1)a = p^(-1)(pbp^(-1))
2. Simplify: p^(-1)a = bp^(-1)
3. Multiplying both sides by the inverse of p^(-1) on the right: (p^(-1)a)p = b
So, b = (p^(-1)a)p.

Given that p is invertible and a = pbp^(-1), we want to solve for b in terms of a.
First, let's multiply both sides of the equation by p:
ap = pb

Now, we can substitute pb with ap from the given equation:
a = apbp^(-1)

Multiplying both sides by p:
ap = apbp^(-1)p
ap = ab

Dividing both sides by a:
b = p^(-1)
Therefore, b is equal to the inverse of p.
In conclusion, b = p^(-1) in terms of a.

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Answer:

w^2-4v^2

Step-by-step explanation:

w*w=w^2

w*2v=2vw

-2v*w=-2vw

-2v*2v=-4v^2

w^2+2vw-2vw-4v^2

w^2-4v^2

The total percentage decrease in the investor's stock account is 13.6%

Percentage is a way of expressing a number as a fraction of 100. It is denoted using the symbol "%". For example, 25% is the same as 25/100 or 0.25.

To find the total percentage decrease in the investor's stock account, we need to first calculate the new values of the stocks after the decreases and then find the percentage decrease of the total value compared to the original value.

The new value of the stock in Company A is:

5750 - 0.16 * 5750 = 4830

The new value of the stock in Company B is:

1200 - 0.02 * 1200 = 1176

The total value of the stocks after the decreases is:

4830 + 1176 = 6006

The percentage decrease of the total value compared to the original value is:

(1 - 6006/6950) * 100% = 13.6%

Therefore, the total percentage decrease in the investor's stock account is 13.6% (rounded to the nearest tenth).

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The degrees of freedom for the sample variance can vary between - [infinity] and + [infinity]. This means that the number of degrees of freedom is not dependent on the sample size, but rather on the amount of variance in the data.

The degrees of freedom for sample variance A. is equal to the sample size minus 1. This means that the correct answer is not provided in your given options. To clarify, let's define the terms:

1. Degrees of freedom: The number of independent values in a statistical calculation that are free to vary.
2. Variance: A measure of dispersion that represents the average squared difference between the values in a dataset and the mean of the dataset.
3. Sample size: The number of observations in a sample.

As the variance increases, the degrees of freedom decrease, which can impact the accuracy of the results. However, it is important to note that a larger sample size can often lead to a more accurate estimate of the population variance, even if the degrees of freedom are not directly related to the sample size.

When calculating the sample variance, the degrees of freedom is equal to the sample size (n) minus 1, often denoted as (n-1). This is because we lose one degree of freedom when estimating the population means using the sample mean.

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The expected value of V is 1.75

The variance of V is 97.65

E[V³]  is 193.083

To find the expected value and variance of V, we first need to find the distribution function of V. For -5 ≤ v < 7, the cumulative distribution function (CDF) F_V(v) can be found by integrating f_V(v):

F_V(v) = ∫ f_V(t) dt

           = ∫ (t+1)²/144 dt

           = (1/144) * ∫ (t² + 2t + 1) dt

           = (1/144) * [(t³)/3 + t² + t]_(-5)^(v)

           = (1/144) * [(v³)/3 + v² + v + 160]/3

For v ≥ 7, F_V(v) = 1.

V's expected value is:

E[V] = ∫ v f_V(v) dv = ∫ (-∞ to -5) 0 dv + ∫ (-5 to 7) v*(v+1)²/144 dv + ∫ (7 to ∞) v dv

= (1/144) * ∫ (-5 to 7) (v³ + v²) dv + ∫ (7 to ∞) v dv

= (1/144) * [(7⁴ - (-5)⁴)/4 + (7³ - (-5)³)/3 + 7²*(7-(-5))]

= 1.75

V's variance is as follows:

Var[V] = E[V²] - (E[V])²

        = ∫ v² f_V(v) dv - (E[V])²

        = ∫ (-∞ to -5) 0 dv + ∫ (-5 to 7) v²*(v+1)²/144 dv + ∫ (7 to ∞) v² dv - (E[V])²

        = (1/144) * ∫ (-5 to 7) (v⁴ + 2v³ + v²) dv + ∫ (7 to ∞) v² dv - (E[V])²

        = (1/144) * [(7⁵ - (-5)⁵)/5 + 2*(7⁴ - (-5)⁴)/4 + (7³3 - (-5)³)/3 + 7*(7²*(7-(-5))) - (1.75)²]

         = 97.65

Finally, we can find E[V³] using:

E[V³] = ∫ v³ f_V(v) dv

          = ∫ (-∞ to -5) 0 dv + ∫ (-5 to 7) v³*(v+1)²/144 dv + ∫ (7 to ∞) v³ dv

          = (1/144) * ∫ (-5 to 7) (v⁵ + 2v⁴ + v³) dv + ∫ (7 to ∞) v³ dv

          = (1/144) * [(7⁶ - (-5)⁶)/6 + 2*(7⁵ - (-5)⁵)/5 + (7⁴ - (-5)⁴)/4 + 7*(7³ - (-5)³)/3]

          = 193.083

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Answer:

Use the graphing calculator to plot the points and then generate the least squares regression line.

y = 1.87837878 - .4594594595

You need a sample size of approximately 10,875 to construct a 90% confidence interval with a margin of error of 0.4 and a standard deviation of 36.

To determine the sample size (n) needed to construct a 90% confidence interval for estimating the population mean, given a standard deviation (σ) of 36 and a margin of error of 0.4, you can use the formula:

[tex]n = (Z * σ / E)^2[/tex]

where:
n = sample size
Z = Z-score corresponding to the desired confidence level (90%)
σ = standard deviation (36)
E = margin of error (0.4)

For a 90% confidence interval, the Z-score is 1.645. Now, plug in the values into the formula:

n = (1.645 * 36 / 0.4)^2
n ≈ 10874.09

Since sample size should be a whole number, round up to the nearest whole number: n ≈ 10875.

So, you need a sample size of approximately 10,875 to construct a 90% confidence interval with a margin of error of 0.4 and a standard deviation of 36.

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Answer:

$164

Step-by-step explanation:

you take the clerk's income ($832.00) and subtract it with the total expenses ($668) to get the savings

The value of p(0 ≤ z ≤ 1.37) is approximately 0.41 (rounded to 2 decimal places).

What is Probability ?

Probability is a branch of mathematics that deals with the study of the likelihood or chance of an event occurring. It is the measure of the likelihood that a particular event or set of events will occur.

To find the value of p(0 ≤ z ≤ 1.37) using Appendix C-1, we need to locate the row containing 0.1 in the far-left column and the column containing 0.37 in the top row.

Starting with the row containing 0.1 in the far-left column, we can locate the value closest to 1.3 in the row, which is 1.37. Moving along the row to the right, we can find the corresponding value of the cumulative distribution function (CDF) for this value of z, which is 0.9147.

Next, we need to find the column containing 0.37 in the top row. The closest value in the column is 0.3707. Moving down the column to the row containing the CDF value we just found, we can read off the value of the CDF for z = 0, which is 0.5000.

To find the value of p(0 ≤ z ≤ 1.37), we subtract the CDF value for z = 0 from the CDF value for z = 1.37:

p(0 ≤ z ≤ 1.37) = 0.9147 - 0.5000 = 0.4147

Therefore, the value of p(0 ≤ z ≤ 1.37) is approximately 0.41 (rounded to 2 decimal places).

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Answer:

y = -1/2(x + 6)^2 - 6

Step-by-step explanation:

0_0

The equilibrium value of spending is 72.

To compute the equilibrium value of spending, we need to use the equation for the expenditure approach to GDP:
GDP = C + I + G + NX

Where:
C = consumption
I = investment
G = government spending
NX = net exports

Given the values of c, i, g, and nx, we can substitute them into the equation:

GDP = 2.75 × GDP + 3 + 2 + 1

Simplifying the equation, we get:

GDP = 2.75 × GDP + 6

Now, we can solve for GDP:

GDP - 2.75 × GDP = 6

0.25 × GDP = 6

GDP = 24

Therefore, the equilibrium value of spending is:

C + I + G + NX = 2.75 × GDP + 3 + 2 + 1 = 2.75 × 24 + 6 = 72

The equilibrium value of spending is 72.

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Answer: 254.34

Step-by-step explanation:

First we can fine the radius by dividing the diameter (18in) by 2: 18/2=9

Then we can use the formula to find the area of the circle (pi*r^2):9*9*pi=81pi

Finally, approximate pi to 3.14 and multiple: 81*3.14=254.34

Therefore, the answer is 254.34

Answer:1017.36

Step-by-step explanation: