A random sample of 11 graduates of a certain secretarial school typed an average of 83.6 words per minute with a standard deviation of 7.2 words per minute. Assuming a normal distribution for the number of words typed per minute, find a 95% confidence interval for the average number of words typed by all graduates of this school.

The sample mean provides the most accurate estimate of the population variance for this particular dataset, as it is calculated directly from the data.

Using the definitional formula and the sample mean for Data Set A:

First, find the sample mean:

[tex]$\bar{x} = \frac{\sum_{i=1}^{n}x_i}{n} = \frac{223}{14} = 15.93$[/tex]

Next, find the variance:

[tex]$s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n} = \frac{199.71}{14} \approx 14.26$[/tex]

Using the definitional formula and a mean score of 15:

[tex]$s^2 = \frac{\sum_{i=1}^{n}(x_i - 15)^2}{n} = \frac{210.93}{14} \approx 15.06$[/tex]

Using the definitional formula and a mean score of 16:

[tex]$s^2 = \frac{\sum_{i=1}^{n}(x_i - 16)^2}{n} = \frac{249.29}{14} \approx 17.81$[/tex]

the results, we can see that the choice of mean score has a significant impact on the variance estimate.

As the mean score increases, the variance estimate also increases. This is because when we use a higher mean score, the deviations from the mean also increase.

This is because when we use a higher mean score, the deviations from the mean also increase.

The sample mean provides the most accurate estimate of the population variance for this particular dataset, as it is calculated directly from the data.

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Answer:

A

Step-by-step explanation:

I am positive that the answer is A) □♡ based on the pattern observed in the given sequence. The symbol □ represents the tens digit and ♡ represents the units digit, and since the previous number in the sequence had ♡ as both digits, the next number should have □ as both digits. Therefore, the next number in the sequence would be □♡, which is a two-digit number where the tens digit is one less than the units digit.

B) □□: This option cannot be the next number in the sequence because it represents a two-digit number where both digits are equal, but the previous number in the sequence had ♡ as both digits. Therefore, the next number should have □ as both digits.

C) ♡♡: This option cannot be the next number in the sequence because it represents a two-digit number where both digits are equal, but the previous number in the sequence had ♡ as both digits. Therefore, the next number should have □ as both digits.

D) ♢□: This option cannot be the next number in the sequence because it represents a two-digit number where the tens digit is greater than the units digit, but the previous numbers in the sequence had the units digit greater than the tens digit. Therefore, the next number should have the tens digit one less than the units digit.

E) ♡♢: This option cannot be the next number in the sequence because it represents a two-digit number where the tens digit is less than the units digit, but the previous numbers in the sequence had the units digit greater than the tens digit. Therefore, the next number should have the tens digit one less than the units digit.

Answer:

Step-by-step explanation:

Because the first digit changes in the first 2 numbers, we can assume that the first number is at the end of a count: like 19, 29, 39...  

So the numbers will be 19,20, 21     or 29, 30 31 etc.

But we know that the last number's second digit is the same as the first number's first first digit, therefore, the square is 1

So the numbers are 19, 20, 21

So the heart will be the first digit of the of the next number, it has to be the 3rd or 5th answer. but we know the next number is 22 so it's the double heart which is the 3rd answer.

Heart heart is answer.

The outward flux of the given vector field across the surface S formed by planes x=0, x=5, y=0, y=25, z=0, and z=125 is 31(5⁸)/2.

The flux of a vector field F across a closed surface S is given by the surface integral of the dot product of F and the unit normal vector to S, which is oriented outward.

In this problem, we need to find the outward flux of the vector field F = 5(xyi + yzj + xzk) across the surface S formed by the planes x=0, x=5, y=0, y=25, z=0 and z=125 in the first octant.

To find the outward normal vector to each of the six surfaces of S, we can use the unit vectors i, j, and k.

For example, the outward normal vector to the plane x=0 is -i, since the plane is perpendicular to the x-axis and points in the negative x direction. Similarly, the outward normal vector to the plane x=5 is i, and so on.

Next, we need to compute the surface area of each of the six planes. The area of the plane

x=5 is (25)(125) = 3125,

and the area of each of the other planes is zero, since they lie on one of the coordinate planes. Therefore, the total surface area of S is

5(3125) = 15,625.

Using the dot product between F and the outward normal vector to each plane, we can find the flux through each plane. The flux through the planes x=0 and x=5 is zero, since the normal vectors are perpendicular to the x component of F.

The flux through the planes y=0 and y=25 is zero, since the normal vectors are perpendicular to the y component of F. The flux through the planes z=0 and z=125 is 5(125)(25), since the normal vectors point in the direction of the z component of F.

Finally, we can add up the flux through each of the six planes to find the total outward flux across S

flux = 2(5)(125)(25) = 31(5⁸)/2

Therefore, the answer is 31(5⁸)/2.The flux of a vector field F across a closed surface S is given by the surface integral of the dot product of F and the unit normal vector to S, which is oriented outward.

Therefore, the answer is 31(5⁸)/2. The correct option is A).

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The table is completed as follows:

Velocity is given by the change in the distance divided by the change in the time, hence the following equation is built to model the relationship between these three variables:

v = d/t.

The velocity for this problem is of 54 miles per hour, hence the distance equation is given as follows:

d = 54t.

For each time, the distances are given as follows:

The time is given as follows:

t = d/54.

For each distance, the times are given as follows:

The table is given by the image presented at the end of the answer.

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The probability that a randomly selected lemur has a tail shorter than 1.7 feet is 0.106. Option E

To solve this problem, we need to standardize the given value of 1.7 feet using the formula:
z = (x - μ) / σ
where x is the given value, μ is the mean, and σ is the standard deviation.
Substituting the values, we get:
z = (1.7 - 1.95) / 0.2
z = -1.25
Now, we need to find the probability of a randomly selected lemur having a tail shorter than 1.7 feet, which is equivalent to finding the area under the standard normal curve to the left of z = -1.25.
Using a standard normal distribution table or calculator, we can find this probability to be approximately 0.106.
Therefore, the answer is option E: 0.106.

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It can be easily verified that the CDF F(x) derived above satisfies all these properties, and hence it is a valid CDF.

To find the cumulative distribution function (CDF) of the random variable X, we integrate the probability density function (PDF) g(x) over the range (-∞, x].

For x < 0, P(X ≤ x) = 0 because the range of X is 0 ≤ X ≤ 1.

For 0 ≤ x ≤ 1, we have:

P(X ≤ x) = ∫[0,x] g(t) dt

P(X ≤ x) = ∫[0,x] (2 - t) dt

P(X ≤ x) = [2t - ([tex]t^2[/tex])/2] evaluated from 0 to x

P(X ≤ x) = 2x - [tex]x^2[/tex]/2

For x > 1, P(X ≤ x) = 1 because the range of X is 0 ≤ X ≤ 1.

Therefore, the CDF of the random variable X is:

F(x) = 0 for x < 0

F(x) = 2x - [tex]x^2[/tex]/2 for 0 ≤ x ≤ 1

F(x) = 1 for x > 1

To check that this is a valid CDF, we need to verify that it satisfies the following properties:

F(x) is non-negative for all x.

F(x) is non-decreasing for all x.

F(x) approaches 0 as x approaches negative infinity.

F(x) approaches 1 as x approaches positive infinity.

It can be easily verified that the CDF F(x) derived above satisfies all these properties, and hence it is a valid CDF.

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Full Question ;

If the probability density of a random variable is given by x g(x)= {2-X 0 0 < x <1 1<x<2 elsewhere Compute u and o

a) 3x + 40C = 4(x + 40)

b) The domain is x >= 0.

c) This expression approaches 60/13, or approximately 46.2%.

(a) We know that the amount of brine in the final mixture is the sum of

the amount of brine in the initial solution and the amount of brine added,

and the total volume of the final mixture is the sum of the initial volume

and the volume added. Therefore,

Amount of brine in final mixture = 0.25(40) + 0.75x

Total volume of final mixture = 40 + x

The concentration of brine in the final mixture is the ratio of the amount

of brine to the total volume of the final mixture. Therefore,

C = (0.25(40) + 0.75x)/(40 + x)

Multiplying numerator and denominator by 4, we get:

4C = (40 + 3x)/ (40 + x)

Simplifying and rearranging, we get:

3x + 40C = 4(x + 40)

(b) The domain of the function is the set of all possible values of x that

make physical sense. Since we cannot add a negative volume of the

75% brine solution, the domain is x >= 0.

(c) The graph of the function is shown below. As the tank is filled, the

rate at which the concentration of brine increases slows down.

As x approaches infinity, the concentration of brine approaches 60%. To

see this, note that as x gets very large, the additional volume of the 75%

brine solution becomes negligible compared to the initial volume of the

tank.

Therefore, the concentration of brine approaches the concentration of

the initial solution, which is 0.25(40)/1000 = 1/25 = 0.04, or 4%. However,

as x approaches infinity, the concentration of brine approaches the

concentration of the 75% brine solution, which is 0.75.

Therefore, the concentration of brine appears to approach the weighted

average of these two concentrations, which is:

0.04(1 - 3x/(40 + 3x)) + 0.75(3x/(40 + 3x))

Simplifying, we get:

(30x + 1600)/(40 + 3x)

As x approaches infinity, this expression approaches 60/13, or

approximately 46.2%.

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The poll's results indicated that approximately 8.5% of individuals between the ages of eight and eighteen could be classified as video game addicts.

This percentage suggests that a significant number of young people are struggling with excessive video game use, which can have negative consequences for their mental and physical health, academic performance, and social relationships.

It's important to note that not all video game use is harmful or addictive. Many individuals enjoy playing video games in moderation, and some even use them as a way to connect with friends and family or improve their cognitive skills.

According to the pie chart the resulting percentage is 8.5%.

However, excessive video game use can lead to addiction, which can be challenging to overcome.

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Fundamental Theorem of Differentiation value of g'(x) when g(x) = Sx 0 √6-5t²dt is -x / (6 - x²) + ∫[0 to √6-x²] x / [tex](6 - t^2)^{(3/2)}[/tex] dt.

To determine g'(x), we need to find the derivative of g(x) with respect to x.

g(x) = ∫[0 to √6-x²] √(6 - t²) dt

Let's use the Fundamental Theorem of Calculus to differentiate g(x):

g'(x) = d/dx [∫[0 to √6-x²] √(6 - t²) dt]

Using the Chain Rule, we can write:

g'(x) = (√(6 - x²))' × √(6 - x²)' - 0

Now, we need to find the derivatives of √(6 - x²) and √(6 - x²)':

√(6 - x²)' = -x / √(6 - x²)

√(6 - x²)" = [tex]-(x^2 + 6 - x^2)^{(-3/2)}[/tex] * (-2x)

Simplifying, we get:

√(6 - x²)' = -x / √(6 - x²)

√(6 - x²)" = x / [tex](6 - t^2)^{(3/2)}[/tex]

Substituting these values, we get:

g'(x) = [(-x / √(6 - x²)) × √(6 - x²)] - ∫[0 to √6-x²] × / [tex](6 - t^2)^{(3/2)}[/tex] dt

Simplifying, we get:

g'(x) = -x / (6 - x²) + ∫[0 to √6-x²] × / [tex](6 - t^2)^{(3/2)}[/tex] dt

Therefore, g'(x) = -x / (6 - x²) + ∫[0 to √6-x²] × / [tex](6 - t^2)^{(3/2)}[/tex] dt.

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The angle measures of the triangles before and after dilation are the same

Given that, we have a triangle that is dilated to form another triangle by a scale factor of n

The dilation transformation is a rigid transformation

This means that it changes the size of a shape after it is applied

However, the shape and the image would be similar shapes and as such would have their angles unchanged

This means that irrespective of the value of the scale factor n, the angle measures would remain the same

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The average number of arrivals per minute at the outpatient clinic is 1/22 or about 0.0455 arrivals according to minute.

If the arrivals at an outpatient clinic follow an exponential distribution with mean 22 mins, then the advent rate, denoted through λ, is identical to 1/22 arrivals in line with minute. that is due to the fact the exponential distribution has a memoryless property, which means that the possibility of an arrival in a given time interval is consistent, and is determined completely via the mean arrival price.

The average number of arrivals according to minute can be calculated using the arrival rate as follows:

Average range of arrivals per minute = λ

Substituting the value of λ, we get:

Average number of arrivals consistent with minute = 1/22

Consequently, At the outpatient clinic, there are about 1/22 arrivals every minute, or around 0.0455 arrivals per minute.

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Here, x = 2 is a vertical asymptote of the function.

Now, For find the vertical asymptotes of the function g(x) = ln(x) / (x - 2), we need to see where the denominator of the fraction is equal to zero, since division by zero is undefined.

Thus, Putting x - 2= 0,

we get, x = 2.

Therefore, x = 2 is a vertical asymptote of the function.

And, We also need to check if there are any other vertical asymptotes.

To do this, we need to check for any values of x that make the numerator of the fraction equal to zero while the denominator is not equal to zero.

However, the numerator ln(x) is never zero for positive values of x, so there are no other vertical asymptotes for this function.

Thus,  x = 2 is a vertical asymptote of the function.

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The value of the integral ∫(i to 0) f(x) dx is -5.

To find the value of the integral, we'll use the properties of definite integrals.

Given that:
∫(i to 1) f(x) dx = -8 (1)
∫(0 to 1) f(x) dx = -3 (2)
We need to find the value of:
∫(i to 0) f(x) dx
Using the properties of definite integrals, we can rewrite the required integral as:
∫(i to 0) f(x) dx = -∫(0 to i) f(x) dx
Now, let's subtract equation (1) from equation (2):
∫(0 to 1) f(x) dx - ∫(i to 1) f(x) dx = -3 - (-8)
This can be simplified as:
∫(0 to i) f(x) dx = 5
Now, we can substitute the value we found into our original equation:
∫(i to 0) f(x) dx = -∫(0 to i) f(x) dx = -5.

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The predicted mean starting salary for a program with a per-year tuition cost of $40,387 is $84,928.

b. Using the least-squares method, we obtain:

bo = -11.075 and by = 2.38

(Note: bo represents the y-intercept, which is the predicted mean starting salary when tuition is 0, and by represents the slope, which is the change in mean starting salary for every unit increase in tuition.)

c. The slope, b, represents the change in mean starting salary for every unit increase in tuition. In this case, the slope is by = 2.38, which means that for every additional $1 in tuition, the mean starting salary upon graduation is expected to increase by $2.38.

Therefore, the correct choice is:

D. For each increase in tuition of $100, the mean starting salary upon graduation is expected to increase by $238.

d. Using the regression equation, we can predict the mean starting salary for a program that has a per-year tuition cost of $40,387:

y = bo + byx

y = -11.075 + 2.38(40,387)

y ≈ $84,928

Therefore, the predicted mean starting salary for a program with a per-year tuition cost of $40,387 is $84,928.

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The distance in meters that was traveled is given as 300 meters

Acceleration = 60 / 40

= 3 / 2

When the velocity that we have in the graph is 30 m/s the time in seconds is twenty seconds

We have to use the formula s = 1 / 2 a t ^2

= 1 / 2 x 3 / 2 x 20^2

= 1200 / 4

= 300 meters

Hence the train traveled for a distance of 300 meters

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The system of linear inequalities represented by the graphs are:  1.6x+2y≤0 and 3x +-2y≤0

You should recall that an inequality is a relationship between two expressions or values that are not equal to each other.

In order to determine the shaded part in an algebraic form of an inequality, like y > 3x + 1, you need to determine if substituting (x, y) into the inequality yields a true statement or a false statement. A true statement means that the ordered pair is a solution to the inequality and the point will be plotted within the shaded region

From the graph the line cuts the x and y axes at 0.6 and 2 for the first line and the region shaded is up

In the second graph the line cuts x and y at 3 and -2 respectively

This gives rise to the solution that the origin is not included Therefore the inequalities formed are 1.6x+2y≤0 and 3x +-2y≤0 respectively

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The five-number summary of the data include the following:

In Mathematics and Statistics, a box plot is sometimes referred to as box-and-whisker plot and it can be defined as a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.

Based on the information provided about the data set, the five-number summary for the given data set include the following:

Minimum (Min) = 36.

First quartile (Q₁) = 38.

Median (Med) = 44.

Third quartile (Q₃) = 50.

Maximum (Max) = 60.

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The predicted value for the top speed of an aircraft if its weight is 100 tons = 359.4679 mph.

There is convincing evidence of a relationship between weight and top speed, assuming the conditions for inference have been met

We are 95% confident that the true slope of the regression line is between 2.871 and 4.085.

Part A)

Independent variable: X: Weight

Response variable: Y: Top speed of an aircraft.

Slope = b = 3.47812

Y-intercept = a = 11.6559

LSRL based on analysis is,

[tex]\hat{Top\ speed}=11.6559+3.47812\ Weight[/tex]

Part B):

Here, p≤ 0.0001 indicates that there linear relationship between two variables. Therefore, we can use LSRL to predict top speed.

Given: Weight = x = 100 tons.

Therefore,

[tex]\hat{Top\ speed}=11.6559+3.47812*100[/tex]

[tex]\hat{Top\ speed}=359.4679\ mph[/tex]

Hence, the predicted value for the top speed of an aircraft if its weight is 100 tons = 359.4679 mph.

Part B:

Conditions for inference:

Linearity: The relationship between weight and top speed is linear.

Independence: The aircrafts' weights and top speeds are independent observations.

Normality: The residuals have a normal distribution.

Equal variance: The residuals have constant variance.

Part C:

To calculate the 95% confidence interval for the slope, we use the formula: slope ± t_critical * s.e. of Coeff.

With 20 degrees of freedom, the t_critical value is approximately 2.086. So, the 95% CI for the slope is: 3.47812 ± (2.086 * 0.294) = (2.871, 4.085).

This means we are 95% confident that the true slope of the regression line is between 2.871 and 4.085.

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Answer:

C is the correct answer

About 92.36% of the population is less than 45 in a normal population with a mean of 35 and a standard deviation of 7.

To find the proportion of the population with a mean (µ) of 35 and a standard deviation (σ) of 7 that is less than 45, follow these steps:

1. Convert the raw score (45) to a z-score using the z-score formula:
  z = (X - µ) / σ
  where X is the raw score (45), µ is the mean (35), and σ is the standard deviation (7).

2. Calculate the z-score:
  z = (45 - 35) / 7
  z ≈ 1.43

3. Use a z-table or calculator to find the proportion of the population corresponding to a z-score of 1.43. The z-table or calculator will provide the area under the curve to the left of the z-score, which represents the proportion of the population that is less than the raw score of 45.

4. The z-table or calculator shows a proportion of approximately 0.9236 for a z-score of 1.43.

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The statement that 'When data with a bell-shaped distribution is standardized, the result will have a standard deviation of 1. However, when data with a wider-spread, bimodal distribution is standardized, the result will tend to have a standard deviation larger than 1' is false.

Standardizing data involves transforming it into a distribution with a mean of 0 and a standard deviation of 1. This is done by subtracting the mean of the original data from each data point and then dividing by the original standard deviation. This process is called z-score calculation.

When data has a bell-shaped distribution, the result of standardization will have a standard deviation of 1, as this is the main goal of standardization. However, when data has a wider-spread, bimodal distribution, the standard deviation of the standardized data will still be 1 after the transformation.

The standardization process ensures that the shape of the original distribution is maintained while changing the mean and standard deviation to the desired values, so regardless of whether the distribution is bell-shaped, bimodal, or any other shape, the standardized data will have a standard deviation of 1.

Hence, the statement is false.

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The mass of the cone candle is volume of empty space = 0.15 cm

The property of a body that is a measure of its inertia and that is commonly taken as a measure of the amount of material it contains and causes it to have weight in a gravitational field

volume of empty space = volume of the cylinder - volume of water

First, we need to calculate the volume of the container

The volume of a cylinder = πr²h

where r = radius of the container   and  

h is the height of the container

r = 3cm  and h = 8 cm  

 π is a constant which is ≈ 3.14

volume of a cylinder = πr²h

                                 =3.14 × 3²×8

                                  =3.14×9×8

                                   =226.08 cm³

We will proceed to find the volume of water

since liquid will take the shape of its container,

then volume of water =  πr²h

r is the radius of the container   and    h is the height of the water

r = 2cm    and   h = 12cm      

volume of water =  πr²h

                            = 3.14 ×2²×12

                             =3.14×4×12

                             =75.56 cm³

volume of empty space = volume of the cylinder - volume of water

                                         =226.08 cm³    -    75.56 cm³ cm³

                                         = 150.52 cm³

Mass = Volume/1000

Mass = 150.52/1000 = 0.15052g

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The volume of the solid of revolution created by rotating the function f(x) = x^2 about the x-axis between x=0 and x=2 is approximately 20.106 cubic units.

The axis of revolution is a line about which a two-dimensional shape is rotated to create a three-dimensional solid. The method for finding the formula to solve for the volume of a solid of revolution depends on the shape being rotated and the axis of revolution.

For example, if we want to find the volume of a solid of revolution created by rotating a function f(x) about the x-axis between the limits of integration a and b, we can use the following formula:

V = π∫[a,b] (f(x))^2 dx

This formula is derived from the shell method, which involves breaking the solid into thin cylindrical shells, finding the volume of each shell, and adding them up. The formula is then the integral of the volume of each shell.

To solve this integral, we can use various methods such as integration by substitution or integration by parts. Once we have found the antiderivative of the integrand, we can evaluate the definite integral using the limits of integration a and b.

For example, if we have the function f(x) = x^2 and we want to find the volume of the solid of revolution created by rotating this function about the x-axis between x=0 and x=2, we can use the formula:

V = π∫[0,2] (x^2)^2 dx

Simplifying this expression, we get:

V = π∫[0,2] x^4 dx

Integrating this expression with respect to x, we get:

V = π[(1/5)x^5] [0,2]

Evaluating this expression at the limits of integration, we get:

V = π[(1/5)(2^5 - 0)]

V = π(32/5)

Therefore, the volume of the solid of revolution created by rotating the function f(x) = x^2 about the x-axis between x=0 and x=2 is approximately 20.106 cubic units.

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Answer:

28.75 inches

Step-by-step explanation:

1.) 5 3/4 x 5/1 = 28 3/4

2.) 28 3/4 simplified is 28.75

Hope that helps!

The probability that at least 9 presentations result in sales is the sum of all these probabilities: P(X ≥ 9) = 0.9392 + 0.8749 + 0.7911 + … + 0 ≈ 0.8732

We can use the normal approximation formula for the Binomial distribution, which states:

μ = np
σ = √(npq)

where n is the number of trials, p is the probability of success, q is the probability of failure (q = 1 - p), μ is the mean, and σ is the standard deviation.

In this case, n = 60, p = 0.2, and q = 0.8. Therefore:

μ = np = 60 x 0.2 = 12
σ = √(npq) = √(60 x 0.2 x 0.8) = 2.19

To find the probability that at least 9 presentations result in sales, we need to find the probability of getting 9, 10, 11, ..., 60 sales, and add them up. However, since the normal distribution is continuous, we need to use a continuity correction by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. This is because the probability of getting exactly 9 (or any other integer) sales is zero, but we want to include the probability of getting between 8.5 and 9.5 sales.

Therefore, the probability that at least 9 presentations result in sales is:

P(X ≥ 9) = P(Z ≥ (8.5 - 12) / 2.19) = P(Z ≥ -1.56) = 0.9392

Similarly, we can find the probability of getting at least 10, 11, …, 60 sales:

P(X ≥ 10) = P(Z ≥ (9.5 - 12) / 2.19) = P(Z ≥ -1.13) = 0.8749
P(X ≥ 11) = P(Z ≥ (10.5 - 12) / 2.19) = P(Z ≥ -0.81) = 0.7911
…  
P(X ≥ 60) = P(Z ≥ (59.5 - 12) / 2.19) = P(Z ≥ 20.29) ≈ 0

The probability that at least 9 presentations result in sales is the sum of all these probabilities:

P(X ≥ 9) = 0.9392 + 0.8749 + 0.7911 + … + 0 ≈ 0.8732

Therefore, the answer is e. 0.8732.

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(a) The derivative value of h′(2) = -35

(b) The derivative value of h′(2) = 8

(c) The derivative value of h′(2) = (-32/16) - (20/16) = -3/2

(d) The derivative value of h′(2) = (2)(-3)(4) + (2)(-3)(-4) + (2)(-2)(4) = -8

(e) The derivative value of h′(2) = (-1)(7)/(2+(-3))² = -7/25

(f) The derivative value of h′(2) = (3/2)(1/2)(7) = 21/4

(g) The derivative value of h′(2) = f′(2g(2))g′(2) = f′(8)(7) = 14

(a) Using the linear properties of the derivative, h′(2) = 5f′(2) - 4g′(2) = -35.

(b) Using the product rule, h′(2) = f′(2)g(2) + f(2)g′(2) = (2)(4) + (-3)(7) = 8.

(c) Using the quotient rule, h′(2) = (g(2)f′(2) - f(2)g′(2)) / g(2)² = (-32/16) - (20/16) = -3/2.

(d) Using the product rule and the chain rule, h′(2) = g(2)f(2) + 2g(2)f′(2) = (-3)(4) + 2(4)(-2) = -8.

(e) Using the quotient rule and the chain rule, h′(2) = -g(2)/(2+(-3))² = -7/25.

(f) Using the chain rule, h′(2) = (1/2)(4 + 3√g(2))g′(2) = (3/2)(1/2)(7) = 21/4.

(g) Using the chain rule, h′(2) = f′(2g(2))g′(2) = f′(8)(7) = 14.

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The length of the diagonal pathway in feet is 8.33. The solution has been obtained by using the Pythagoras theorem.

What is Pythagoras theorem?

Pythagoras' Theorem states that the square of a right-angled triangle's hypotenuse side is equal to the sum of the squares of its other two sides.

We are given that the dimensions of the rectangle are 60‘ x 80‘.

This means that the perpendicular is 60 inches and base is 80 inches.

Let the diagonal pathway be 'H'.

So, using the Pythagoras theorem, we get

⇒ [tex]60^{2}[/tex] + [tex]80^{2}[/tex] = [tex]H^{2}[/tex]

⇒ 3600 + 6400 = [tex]H^{2}[/tex]

⇒ 10000 = [tex]H^{2}[/tex]

⇒ H = 100 inches

We know that 1 foot = 12 inches.

So,

100 inches = 8.33 feet

Hence, the length of the diagonal pathway in feet is 8.33.

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The critical point is  (B/12, 1/12) and  f(x,y) has a saddle point at (B/12, 1/12).

To find the critical points of f(x,y), we need to find where the partial derivatives with respect to x and y are both zero:

∂f/∂x = 1 - 12y = 0

∂f/∂y = -12x + B = 0

From the first equation, we have y = 1/12.

Substituting into the second equation, we get:

-12x + B = 0

⇒ x = B/12

So the critical point of f(x,y) is (B/12, 1/12).

To find the relative extrema and saddle points, we need to use the second partial derivative test. We have:

∂²f/∂x² = 0 (constant)

∂²f/∂y² = 0 (constant)

∂²f/∂x∂y = -12 (constant)

At the critical point (B/12, 1/12), the determinant of the Hessian matrix is:

∂²f/∂x²× ∂²f/∂y² - (∂²f/∂x∂y)² = 0× 0 - (-12)² = 144

Hence, the determinant is positive and ∂²f/∂x² is zero, we can conclude that f(x,y) has a saddle point at (B/12, 1/12).

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