What are the sine and cosine ratios for a 30°–60°–90° triangle with a hypotenuse of 36, when θ = 60°?

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 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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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 correct interpretation of the slope in the context of the data is that for every additional year a professor works at a college, his/her salary is predicted to increase by $1,280.

The given regression equation for professor salaries is Salary = 95000 + 1280 (Years), where "Years" represents the number of years a professor has worked at a college, and "Salary" represents the annual salary (in dollars) the professor earns. The slope of 1280 in the regression equation represents the change in Salary for each unit increase in Years.

Therefore, for every additional year a professor works at a college, his/her salary is predicted to increase by $1,280. This means that as a professor gains more experience and works for more years at a college, their salary is expected to increase by $1,280 per year, according to the given regression equation.

Therefore, the correct interpretation of the slope is: For every additional year a professor works at a college, his/her salary is predicted to increase by $1,280.

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The required z-test statistic is 2.45 and p-value for this test is approximately 0.014

Given,

Population mean waiting time for service, μ = 6 minutes

Sample size, n = 60

Sample mean waiting time for service is, M = 6.5 minutes

Population standard deviation is, σ = 1.9 minutes

To determine the p-value for this test.

Since it is claimed that customers spend an average of 6 minutes waiting for service at the store's deli counter; therefore, the appropriate null and the alternate hypothesis are:

H0:μ=6

Ha:μ≠6

This corresponds to a two-tailed test.

Since the population standard deviation is known; therefore, the z-test is appropriate.

Assuming the null hypothesis true, the calculated z-value is obtained as:

z=M−μ/(σ/√n)=6.5−6/(1.9/√60)≈2.446

Therefore, the required calculated z-value is approximately 2.45.

The two-tailed p-value corresponding to the z-value of 2.45 is:

p−value≈ .014286

Therefore, the required p-value for this test is approximately 0.014

(rounded to three decimals).

The result is significant at p < 0.10.

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Answer:

15×(-1)^10 -2*(-1)-3 = 14

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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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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The function used in Excel to find the "line of best fit" is called "Trendline". To add a trendline to a chart, you can right-click on a data series in the chart and select "Add Trendline" from the drop-down menu.

A trendline is a line that shows the general pattern or direction of a set of data. It's also known as a line of stylish fit or a retrogression line. A trendline can be added to a map in Excel to help  fantasize the relationship between two variables and to make  prognostications grounded on the data.  

When you add a trendline in Excel, you have the option to choose from several different types of retrogression models,  similar as direct, exponential, logarithmic, polynomial, power, and moving average. Each type of model fits a different type of data pattern, and it's important to choose the applicable model for your data.   In addition to adding a trendline, Excel also provides a residual plot to help you assess the  virtuousness of fit of the trendline.

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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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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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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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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 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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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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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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(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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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

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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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 integral ∫(0 to 4) 1/(sqrt(x)(1+sqrt(x))) dx is equal to 2 ln(3), which corresponds to option C.

To determine whether the integral ∫(0 to 4) 1/(sqrt(x)(1+sqrt(x))) dx is divergent or convergent, and find its value, follow these steps:

Step 1: Make a substitution
Let u = sqrt(x), so x = u^2 and dx = 2u du.
The integral now becomes:
∫(0 to 2) 1/(u(1+u)) * 2u du

Step 2: Simplify the integral
The integral simplifies to:
∫(0 to 2) 2/(1+u) du

Step 3: Integrate the function
Integrate the simplified function with respect to u:
∫(0 to 2) 2/(1+u) du = 2 ∫(0 to 2) 1/(1+u) du = 2[ln(1+u)](0 to 2)

Step 4: Evaluate the definite integral
Evaluate the definite integral using the limits:
2[ln(1+2) - ln(1+0)] = 2[ln(3) - ln(1)] = 2(ln(3) - 0) = 2 ln(3)

So, the integral ∫(0 to 4) 1/(sqrt(x)(1+sqrt(x))) dx is equal to 2 ln(3), which corresponds to option C.

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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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The square inches of paint that will be needed for the sides is 3858.45 sq in

From the question, we have the following parameters that can be used in our computation:

The pentagonal prism

The square inches of paint that will be needed for the sides is the surface area of the pentagonal, and this is calculated as

Area = 5ah + 1/2√[5(5+2√5)]a²

Where

a = side length = 27 inches

h = height = 10 inches

Substitute the known values in the above equation, so, we have the following representation

Area = 5 * 27 * 10 + 1/2√[5(5+2√5)] * 27²

Evaluate

Area = 3858.45

Hence, the square inches of paint that will be needed for the sides is 3858.45 sq in

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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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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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(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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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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