Question 16 (2 points) In order to say that something like taxi accidents are caused by drivers wearing heavy coats, there needs to be a Pearson's correlation coefficient r of at least 0.9 between them. True False

If a = 17.5, quadrilateral DEFG will be a parallelogram.

What is quadrilateral?

A quadrilateral is a geometric shape that has four sides and four vertices (corners). The angles formed by the sides of a quadrilateral add up to 360 degrees. Some common examples of quadrilaterals include squares, rectangles, parallelograms, trapezoids, and kites.

For a quadrilateral to be a parallelogram, opposite sides must be parallel.

Therefore, EF || DG and DE || FG.

Since EF and DG are both horizontal, they must have the same y-coordinate.

So, EF = DG = 18.

Also, DE and FG are both vertical, so they must have the same x-coordinate.

So, FG = DE = 2a - 17.

Since DE and FG are equal, we have:

2a - 17 = 18

Solving for a, we get:

a = 17.5

Therefore, if a = 17.5, quadrilateral DEFG will be a parallelogram.

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The value of the integral is 1710. To evaluate the integral S1 0 (1+1/2u⁴ - 2/5u⁹)du, we need to integrate each term separately.

∫1du = u + C, where C is the constant of integration.

To integrate 1/2u⁴, we can use the power rule of integration:

∫1/2u⁴ du = (1/2) ∫u⁴ du = (1/2) * u⁵/5 + C = u⁵/10 + C

To integrate -2/5u⁹, we can also use the power rule of integration:

∫(-2/5)u⁹ du = (-2/5) ∫u⁹ du = (-2/5) * u¹⁰/10 + C = -u¹⁰/25 + C

Putting everything together, we have:

∫(1+1/2u⁴ - 2/5u⁹)du = ∫1du + ∫1/2u⁴ du - ∫2/5u⁹ du

= u + u⁵/10 - (-u¹⁰/25) + C

= u + u⁵/10 + u¹⁰/25 + C

Now, we can evaluate the definite integral by plugging in the limits of integration:

S1 0 (1+1/2u⁴ - 2/5u⁹)du = [u + u⁵/10 + u¹⁰/25]₁⁰

= (10 + 10⁵/10 + 10¹⁰/25) - (0 + 0⁵/10 + 0¹⁰/25)

= 10 + 1000 + 400000/25

= 10 + 1000 + 16000

= 1710

Therefore, the value of the integral is 1710.

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To find the absolute extreme values of the function f(x) = 2[tex]x^{3}[/tex]- 24 on the interval [-3, 3], follow these steps:

1. Find the critical points by taking the derivative of the function and setting it to zero.
f'(x) = 6[tex]x^{2}[/tex]

2. Solve for x:
6[tex]x^{2}[/tex] = 0
x = 0

3. Evaluate the function at the critical point and the interval's endpoints:
f(-3) = 2[tex](-3)^{3}[/tex]- 24 = -90
f(0) = 2[tex](0)^{3}[/tex]- 24 = -24
f(3) = 2[tex](3)^{3}[/tex] - 24 = 90

4. Compare the function values and identify the absolute extremes:
Absolute minimum: f(-3) = -90
Absolute maximum: f(3) = 90

So, the absolute extreme values of the function f(x) = 2[tex]x^{3}[/tex] - 24 on the interval [-3, 3] are -90 (minimum) and 90 (maximum).

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We can say with 99% confidence that the true proportion of people who like dogs is within the range of 6% +/- 5% (i.e., between 1% and 11%).

The margin of error (MOE) is a measure of how much the results of a survey may differ from the true population values. It is affected by the sample size and the level of confidence of the survey.

To calculate the margin of error at the 99% confidence level for this poll, we can use the following formula:

MOE = z * (sqrt(p*q/n))

where:

z is the z-score corresponding to the confidence level. For a 99% confidence level, z = 2.576

p is the proportion of respondents who said they liked dogs, which is 0.06 in this case

q is the complement of p, which is 1 - 0.06 = 0.94

n is the sample size, which is 150 in this case

Plugging in the values, we get:

MOE = 2.576 * (sqrt(0.06*0.94/150)) = 0.049

Rounding to three decimal places, the margin of error is 0.049 or approximately 0.05.

Therefore, we can say with 99% confidence that the true proportion of people who like dogs is within the range of 6% +/- 5% (i.e., between 1% and 11%).

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Answer:

Step-by-step explanation:

For a system of equations, the solution is where the 2 lines intersect. They intersect at (-3,1).  But they only wan the y-coordinate, so it's the y part of the answer (x,y) x=-3  and y=1

So your answer is 1

Answer:

500

Step-by-step explanation:

i think. i have down syndrome dont hate my answers.

Answer: 500

Step-by-step explanation:

The probability of having fewer than 15 no-shows among 100 advanced reservations is approximately 0.8508

In our case, np = 100 * 0.12 = 12 and n(1-p) = 100 * 0.88 = 88, so we meet the criteria for using the normal approximation.

Next, we'll use the normal distribution formula to find the probability that there will be fewer than 15 no-shows:

P(X < 15) = P(Z < (15 - 12) / 2.60) = P(Z < 1.15)

Here, Z is a standard normal variable with mean 0 and standard deviation 1. We can use a normal distribution table or calculator to find that P(Z < 1.15) is approximately 0.8749.

However, we need to include the correction for continuity since we're approximating a discrete binomial distribution with a continuous normal distribution.

The correction for continuity involves adjusting the boundaries of the interval by 0.5. In this case, we're interested in the probability of having fewer than 15 no-shows, so we'll adjust the upper boundary to 14.5:

P(X < 15) ≈ P(Z < (14.5 - 12) / 2.60) = P(Z < 1.04)

Using a normal distribution table or calculator, we can find that P(Z < 1.04) is approximately 0.8508.

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A strategy to find the total area of water that will be sprayed by the four nozzles is to first find the total arc measure covered by the four nozzles and then find the fraction of the circle covered by the sprayed water.

The sum arc measure of the water spray in each sector, produced by all four nozzles, totals to 100 degrees. To calculate what portion of the circular fountain is covered by the sprayed water, divide this value by the circle's 360-degree total. For convenience, let us define r as the radius of the fountain. Then find the area of the circle.

Next, expand your knowledge of the coverage area further and multiply the fraction by the entire circular fountain's span to find the precise square footage. In turn, determining the wet sides' length requires assessing the circumference of the entire structure and multiplying it again by the fractional width measured earlier from the nozzle spray.

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The lines L1 and L2 will intersect at an intersection point (-1,3,-4) for L1 is (-1,1,-2), and the directional vector of L2 is (1,1,3).

We need to compare their directional vectors to determine the relationship between L1 and L2. The directional vector of L1 is (-1,1,-2), and the directional vector of L2 is (1,1,3).

Since these vectors are not scalar multiples of each other, the lines are not parallel.

To determine if they intersect or are skew, we can find the point of intersection using the system of equations formed by setting the equations of L1 and L2 equal to each other.

Solving this system of equations, we find that x = -1, y = 3, and z = -4.

Therefore, the lines intersect at the point (-1,3,-4).

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The acceleration vector a(t) is (-9 cos 3t)i + (-2 sin t)j.

If r(t) is the position vector of a particle in the plane at time t, and r(t) = (cos 3t) i + (2 sin t) j, you want to find the acceleration vector.

First, find the velocity vector by taking the derivative of the position vector with respect to time:

v(t) = dr(t)/dt = (-3 sin 3t) i + (2 cos t) j

Next, find the acceleration vector by taking the derivative of the velocity vector with respect to time:

a(t) = dv(t)/dt = (-9 cos 3t) i + (-2 sin t) j

The acceleration vector a(t) is (-9 cos 3t)i + (-2 sin t)j.

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The probability that 3 of 8 stolen cars will be recovered is 0.296 or approximately 0.30.

The given problem involves a binomial distribution, where the probability of success (recovering a stolen car) is p = 0.78 and the number of trials is n = 8. We need to find the probability of getting exactly 3 successes.

The probability of getting exactly k successes in n trials can be calculated using the binomial probability formula:

P(k successes) = (n choose k) * [tex]p^k[/tex] * [tex]{1-p}^{n-k}[/tex]

where (n choose k) represents the binomial coefficient, which can be calculated as:

(n choose k) = n! / (k! * (n-k)!)

where n! represents the factorial of n.

Using the above formula with k = 3, n = 8, and p = 0.78, we get:

P(3 successes) = (8 choose 3) * 0.78³ * (1-0.78)⁵

= 56 * 0.78³ * 0.22⁵

= 0.296

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The length of the rectangle with an area of 140 square meters is equal to 14 meters.

Area of the rectangle = 140square meters

Let us consider the length of the rectangle be L

and the width of the rectangle be W.

The width is 4 meters less than the length, so we can write,

W = L - 4

The area of the rectangle is 140 square meters,

Area of the rectangle = L x W

Substituting the expression for W into the equation for the area, we get,

⇒Area of the rectangle = L x (L - 4)

Now plug in the value of the area and solve for L,

⇒ 140 = L x (L - 4)

⇒ 140 = L^2 - 4L

⇒ L^2 - 4L - 140 = 0

Solve this quadratic equation by factoring or by using the quadratic formula.

⇒ L^2 - 14L + 10L - 140 = 0

⇒(L - 14)(L + 10) = 0

This gives us two possible solutions for L,

L = 14 or L = -10.

Since the length of the rectangle cannot be negative,

Discard the negative solution

And conclude that the length of the rectangle is L = 14 meters.

⇒ width W = L - 4

                  = 14 - 4

                  = 10 meters.

Therefore, the length of the rectangle is equals to 14 meters.

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The set of angles that does NOT produce a triangle is 43°, 62°, and 76°, as their sum is 181°

To determine if the given angles can form a triangle, we need to check if the sum of the three angles is equal to 180° (the sum of the angles of a triangle).

62°, 34°, and 84°:

Sum = 62° + 34° + 84° = 180°

26°, 48°, and 106°:

Sum = 26° + 48° + 106° = 180°

43°, 62°, and 76°:

Sum = 43° + 62° + 76° = 181°

34°, 67°, and 79°:

Sum = 34° + 67° + 79° = 180°

The probability that fewer than 12 of the 21 randomly selected students prefer online exams is 0.0269, or 2.69%.

According to a study, 74% of students prefer online exams to in-class exams. If 21 students are randomly selected, you want to know the probability that fewer than 12 of these students prefer online exams.

To answer this question, we can use the binomial probability formula:

P(x) = C(n, x) × pˣ × (1-p)^(n-x)

where:
- P(x) is the probability of having exactly x successes in n trials
- C(n, x) is the number of combinations of n items taken x at a time
- n is the number of trials (21 students in this case)
- x is the number of successful trials (students preferring online exams)
- p is the probability of success (0.74, the percentage of students preferring online exams)

Since we want the probability of fewer than 12 students preferring online exams, we need to calculate the sum of probabilities for x = 0 to 11:

P(x < 12) = Σ [C(21, x) × 0.74ˣ × (1-0.74)⁽²¹⁻ˣ⁾] for x = 0 to 11

Using a calculator or statistical software to compute the probabilities, the sum of the probabilities for x = 0 to 11 is approximately 0.0269.

So, the probability that fewer than 12 of the 21 randomly selected students prefer online exams is 0.0269, or 2.69%.

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The variance estimate using the sample mean for Data Set A is 20.621.

The variance estimate using a mean score of 15 is 21.238.

The variance estimate using a mean score of 16 is 17.810.

Variance estimates are based on a relatively small sample size of 14, and may not be representative of the true population variance.

To compute variance estimates for Data Set A, we can use the following definitional formula:

[tex]variance = \Sigma(xi - x)^2 / (n - 1)[/tex]where xi is the i-th score in the data set, x is the mean score, and n is the sample size.

Using the sample mean for Data Set A:

First, we need to compute the sample mean x for Data Set A:

[tex]x = (23 + 13 + 13 + 7 + 9 + 19 + 11 + 19 + 15 + 14 + 17 + 21 + 21 + 17) / 14[/tex]

x = 15.1429 (rounded to 4 decimal places)

Compute the variance using the above formula:

[tex]variance = \Sigma(xi - x)^2 / (n - 1)[/tex]

[tex]variance = [(23 - 15.1429)^2 + (13 - 15.1429)^2 + ... + (17 - 15.1429)^2] / (14 - 1)[/tex]

variance = 20.6207 (rounded to 3 decimal places)

The variance estimate using the sample mean for Data Set A is 20.621.

Using a mean score of 15:

If we use a mean score of 15, we can compute the variance using the same formula as above, but with x = 15:

[tex]variance = [(23 - 15)^2 + (13 - 15)^2 + ... + (17 - 15)^2] / (14 - 1)[/tex]

variance = 21.2381 (rounded to 3 decimal places)

The variance estimate using a mean score of 15 is 21.238.

Using a mean score of 16:

A mean score of 16, we can compute the variance using the same formula as above, but with x = 16:

[tex]variance = [(23 - 16)^2 + (13 - 16)^2 + ... + (17 - 16)^2] / (14 - 1)[/tex]

variance = 17.8095 (rounded to 3 decimal places)

The variance estimate using a mean score of 16 is 17.810.

Conclusions:

From the above results, we can see that the variance estimate is sensitive to the choice of the mean score.

As the mean score increases, the variance estimates decrease, and as the mean score decreases, the variance estimates increase.

This is because the variance is a measure of how spread out the data is from the mean score.

The mean score is higher, the data tends to be more tightly clustered around the mean, resulting in a smaller variance estimate.

Conversely, when the mean score is lower, the data tends to be more spread out, resulting in a larger variance estimate.

The variance estimate using the sample mean (20.621) is between the variance estimates using a mean score of 15 (21.238) and a mean score of 16 (17.810).

The sample mean is a reasonable estimate of the population mean, and that the data is not overly skewed in one direction or the other.

Variance estimates are based on a relatively small sample size of 14, and may not be representative of the true population variance.

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Answer:

15×(-1)^10 -2*(-1)-3 = 14

True, a cubic centimeter (cm^3) is a unit of volume.

Volume is the measure of space that an object occupies, and the cubic centimeter is a commonly used unit to express volume. In a cubic centimeter, each side of the cube measures 1 centimeter, and the total volume is 1 centimeter x 1 centimeter x 1 centimeter = 1 cm^3.

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(1) The value of c that makes the equation a perfect square is 196 and the expression in factor form is 4(x - 7)².

(2) The value of x using completing the square method is x = ¹/₄ (19 ± √229).

To make the equation a perfect square trinomial, we need to take half of the coefficient of x and square it, and then add that result to the expression.

4x² - 28x + c

The coefficient of x is -28,

= ¹/₂(-28) =  -14.

(-14)² =  196.

Therefore, the value of c that makes the equation a perfect square trinomial is 196.

So, the expression in factor form is;

4x² - 28x + 196 = 4(x - 7)²

2. The solution of the equation by completing the square method;

4x² - 38x = -33

x² - (38/4)x = -33/4

half of coefficient of x = -38/8, the square = (-38/8)²;

(x - 38/8)² = -33/4 + 361/16

(x - 38/8)² = 229/16

x - 38/8 = ±√ (229/16)

x = ±√229/4 + 38/8

x = ±√229/4 + 19/4

x = ¹/₄ (19 ± √229)

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The total square footage of the two-story home is 7,200 square feet.

To calculate the total square footage of the home, we need to multiply the exterior wall measurements of each floor and add them together.

Assuming both floors have the same dimensions, the square footage of one floor would be:

40 feet x 90 feet = 3,600 square feet

Since the house has two floors, we need to double this number:

3,600 square feet x 2 = 7,200 square feet

Therefore, the total square footage of the two-story home is 7,200 square feet.

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The length will be 375 m.

The width will be 250 m.

What is perimeter?

The complete length of a shape's edge serves as its perimeter in geometric terms. Adding the lengths of all the sides and edges that surround a form yields its perimeter. It is calculated using linear length units such centimeters, meters, inches, and feet.

let the length be x and the width be w

The perimeter will be:

2x+3w=1500

thus

3w=(1500-2x)

w=(1500-2x)/3

w=500-2/3x

The area will be:

A=x*w

A=x(500-2/3x)

A=500x-(2/3)x²

The above is a quadratic equation; thus finding the axis of symmetry we will evaluate for the value of x that will give us maximum area.

Axis of symmetry:

x=-b/(2a)

from our equation:

a=(-2/3) and b=500

thus

x=-500/[2(-2/3)]

x=375

the length will be 375 m

The width will be 250 m

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x = ±√(4t/ arctan(t/4)), we have two solutions for x, because we didn't specify which sign to take when taking the square root. Finally, we can check that this solution satisfies the differential equation and the initial condition.

To solve this differential equation using separation of variables, we need to separate the variables x and t on opposite sides of the equation and integrate each side separately. Here's how we can do it:

(t^2 + 16) dx/dt = x^2 + 16

Dividing both sides by (x^2 + 16), we get:

(t^2 + 16)/(x^2 + 16) dx/dt = 1

Now we can separate the variables:

(x^2 + 16)/(t^2 + 16) dx = dt

Integrating both sides:

∫(x^2 + 16)/(t^2 + 16) dx = ∫dt

To evaluate the integral on the left, we can use the substitution u = t/4, du = 1/4 dt:

∫(x^2 + 16)/(t^2 + 16) dx = 4∫(x^2 + 16)/(16u^2 + 16) dx

= 4∫(x^2 + 16)/(4u^2 + 4) dx

= 4∫(x^2/4 + 4)/(u^2 + 1) dx

= 4(x^2/4 arctan(u) + 4u) + C

= x^2 arctan(t/4) + 16t/4 + C

where C is the constant of integration. Now we can solve for x by plugging in the initial condition x(0) = 4:

x^2 arctan(0/4) + 16(0)/4 + C = 4^2

C = 16

So the particular solution is:

x^2 arctan(t/4) + 16t/4 + 16 = 16 + x^2 arctan(t/4)

Simplifying:

x^2 arctan(t/4) = 4t

x^2 = 4t/ arctan(t/4)

Therefore, x = ±√(4t/ arctan(t/4))

Note that we have two solutions for x, because we didn't specify which sign to take when taking the square root. Finally, we can check that this solution satisfies the differential equation and the initial condition.
To solve the differential equation (t² + 16) dx/dt = (x² + 16) with the initial condition x(0) = 4 using separation of variables, follow these steps:

1. Rewrite the equation as (t² + 16) dx = (x² + 16) dt.
2. Separate the variables: (1/(x² + 16)) dx = (1/(t² + 16)) dt.
3. Integrate both sides: ∫(1/(x² + 16)) dx = ∫(1/(t² + 16)) dt + C.
4. The antiderivatives are: (1/4)arctan(x/4) = (1/4)arctan(t/4) + C.
5. Apply the initial condition x(0) = 4: (1/4)arctan(4/4) = (1/4)arctan(0/4) + C, which simplifies to (1/4)(π/4) = C.
6. Solve for x(t): arctan(x/4) = arctan(t/4) + π.

The solution for x(t) is:
x(t) = 4 * tan(arctan(t/4) + π).

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A random sample 50 employees yielded a mean of 2.79 years that employees stay in the company and o- 76. We test for the nut hypothesis that the population mean is less or equal than.inst the alternative hypothesis that the population mean is greater than 3 At a significance leve 0.01, we have enough evidence that the average time is less than 3 years is true.

To determine whether the null hypothesis (population mean <= 3) can be rejected in favor of the alternative hypothesis (population mean > 3) at a significance level of 0.01, we can conduct a one-sample t-test.

The test statistic is calculated as follows:
t = (sample mean - hypothesized population mean) / (sample standard deviation / sqrt(sample size))
Plugging in the given values, we get:
t = (2.79 - 3) / (0.76 / sqrt(50))
t = -2.12
The degrees of freedom for this test is 49 (sample size - 1). Using a t-distribution table with 49 degrees of freedom and a one-tailed test at a significance level of 0.01, we find a critical value of 2.405. Since our calculated t-value (-2.12) is less than the critical value (-2.405), we can reject the null hypothesis in favor of the alternative hypothesis.

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The marginal profit function at x = 1,500 is P'(1500) = 155 dollars per unit.

To find the marginal profit function, we first need to find the revenue and profit functions using the given price-demand and cost functions.

1. Price-demand function: p = 295 - (x/80)
2. Cost function: C(x) = 36,000 + 110x

First, find the revenue function, R(x). Revenue is the product of the price per unit and the number of units sold, so R(x) = px.

R(x) = (295 - (x/80))x

Next, find the profit function, P(x). Profit is the difference between revenue and cost, so P(x) = R(x) - C(x).

P(x) = (295 - (x/80))x - (36,000 + 110x)

Now, we'll find the derivative of the profit function with respect to x, which is the marginal profit function, P'(x).

P'(x) = d/dx[(295 - (x/80))x - (36,000 + 110x)]

Using the product rule and the constant rule, we get:

P'(x) = (295 - (x/80)) - x/80 + (-110)

Simplify the expression:

P'(x) = 295 - 2x/80 - 110

Now, evaluate the marginal profit function at x = 1,500.

P'(1500) = 295 - 2(1500)/80 - 110

Calculate the result:

P'(1500) = 295 - 30 - 110 = 155

Therefore, the marginal profit function at x = 1,500 is P'(1500) = 155 dollars per unit.

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The probability that a continuous random variable takes on exactly a specific value is 0 because there are an infinite number of possible values that the variable can take on. Option (D) is the correct answer.

For any continuous random variable, the probability that the random variable takes on exactly a specific value is 0. This is because continuous random variables can take on an infinite number of possible values within a given range. As such, the probability of any single specific value occurring is infinitesimally small.

To understand why this is the case, consider a real-life example of measuring the height of a person. A continuous random variable is used to represent the height of a person because height can take on an infinite number of values between any two given values. For instance, if we measure the height of a person to be exactly 5 feet and 10 inches, we know that the true height of the person is not exactly 5 feet and 10 inches. It could be slightly taller or slightly shorter than 5 feet and 10 inches, depending on the precision of the measuring tool used.

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(a) The length of /AC/ is 12 m

(b) The length of /AG/ is 12.5 m

(c) The angle line AG makes with the floor is 16.3°

Length is the distance between two points.

(a) To calculate the length AC of the cuboid, we use the formula below

Formula:

Where:

Substitute these values into equation 1

(b) Similarly, to calculate the value of AG, WE use the formula below

Where:

Substitute these values into equation 2

(c) Finally, to find the angle that AG make to the floor, we use the formula below

Given:

Substitute these values into equation 3

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The height of the cone needs to be 6 inches if it has the same width as the 3-inch wide, 2-inch tall cup to hold the same amount of ice cream

What is Volume of cone?

Volume of a cone = π r² h/3

Volume of cone = 1/3 * π * r² * h

Volume of cylinder = π * r² * h

Volume of cylinder = π * r² * h

π * (1.5)² * 2 = 4.5π

Volume of cone = 1/3 * π * r² * h

4.5π = 1/3 * π * (1.5)² * h

4.5π = 0.75π * h

h = 4.5π / 0.75π

h = 6

Therefore, the height of the cone needs to be 6 inches if it has the same width as the 3-inch wide, 2-inch tall cup to hold the same amount of ice cream

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The area of the square is increasing at a rate of 40 cm²/s when the area is 16 cm².

We are required to determine the rate at which the area of a square is increasing when each side is increasing at 5 cm/s and the area is 16 cm².

First, let's establish some variables:

Let s be the side length of the square
Let A be the area of the square
ds/dt is the rate at which the side length is increasing, which is given as 5 cm/s
dA/dt is the rate at which the area is increasing, which we need to find

The area of a square is given by the formula:

A = s².

Now, we can differentiate both sides with respect to time (t):

dA/dt = 2s * (ds/dt)

We know that the area A is 16 cm². Since A = s², we can find the side length s:

s² = 16

s = 4 cm

Now, plug the values of s and ds/dt into the equation we derived:

dA/dt = 2 * 4 * 5

dA/dt = 40 cm²/s

The rate of change for the area is 40 cm²/s.

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Answer:

60%

Step-by-step explanation:

12 / 20 students did the homework

12 / 20 = 3 / 5 = 0.6

0.6 as a percent is 60%

So, 60% of the class did the homework

The relationship between the number of hours worked and the number of toys built can be represented by a linear equation y = 9x, where y is the number of toys built and x is the number of hours worked. The graph is attached below.

The graph representing this relationship is a straight line passing through the origin (0,0) with a slope of 9. The x-axis represents the number of hours worked, and the y-axis represents the number of toys built. As x increases, y increases proportionally at a rate of 9 units of y for every unit of x.

The slope of the line, which is the ratio of the change in y to the change in x, represents the rate of increase of the number of toys built per hour worked. In this case, the slope is 9, which means that the number of toys built increases by 9 for every additional hour worked.

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We can say with 95% confidence that the true mean life of light bulbs in this shipment falls between 371.84 hours and 448.16 hours.

To construct a 95% confidence interval estimate for the population mean life of light bulbs in this shipment, we can use the following formula:

Confidence interval = sample mean +/- margin of error

where the margin of error is given by:

Margin of error = (critical value) x (standard deviation / sqrt(sample size))

Since we want a 95% confidence interval, the critical value is 1.96 (from the standard normal distribution table). Plugging in the given values, we get:

Margin of error = 1.96 x (108 / sqrt(81)) = 38.16

Therefore, the confidence interval estimate is:

410 +/- 38.16

or

(371.84, 448.16)

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