Question 1

A person in a rowboat 2 miles from the nearest point on a straight shoreline wishes to reach his house which is 6 miles farther down the shore. If he can row at a rate of 6 miles per hour and run at a rate of 10 miles per hour, what is the minimum amount of time (in minutes) it will take him to get home?

Accepted Answer

The person can row 2 miles to the shoreline in $\frac{2}{6} = \frac{1}{3}$ hours or 20 minutes.
Then the person needs to run the remaining distance of $6 - 2 = 4$ miles at a rate of 10 mph, which takes $\frac{4}{10} = \frac{2}{5}$ hours or 24 minutes.
So the total time it takes is 20 + 24 = 44 minutes, which is closest to Choice C, 52 minutes.

Question 2

A Norman window has the outline of a semicircle on top of a rectangle. Suppose there is  feet of wood trim available. Find the dimensions of the rectangle (and, hence, the semicircle) that will maximize the area of the window.

Accepted Answer

Let the rectangle have a base of length x and height y, and let r be the radius of the semicircle. Then the total wood trim used is x+2y+πr. We want to maximize the area of the window, which is xy+πr^2/2. Since we have a constraint on the amount of wood trim used, we can solve for one of the variables in terms of the other and substitute into the area formula. For example, solving for r gives r=(x/2)+y/2. Substituting this into the area formula results in A(x,y)=xy+(π/8)(x^2+4y^2). This is a function of two variables, so we can find critical points by taking the partial derivatives and setting them equal to zero: ∂A/∂x=y+(π/4)x=0 and ∂A/∂y=x+(π/2)y=0. Solving these simultaneous equations gives x=(2/π)y and y=(4/π^2)x. Substituting either of these into the constraint x+2y+πr=11 gives the value for the corresponding variable: x=11/(2+π) and y=11(4π-2π^2)/(π^2+4). Plugging these into the area formula gives A≈8.45. Note that this is a local maximum since the second partial derivatives of A with respect to x and y are both positive. The answer is therefore D, with the given dimensions for x and y.

Question 3

A company's revenue for selling x (thousand) items is given by  in millions of dollars. Find the value of x (rounded to the nearest item) that maximizes the revenue and find the maximum revenue (rounded to the nearest dollar).

Accepted Answer

The answer of A company's revenue for selling x (thousand)...

Question 4

Find the volume of the largest box with no top that can be made from a piece of cardboard 60 inches square by cutting equal squares from the corners and turning up the sides.

Accepted Answer

The volume of the box is given by V = x^3, where x is the length of the side of the square that is cut from each corner. To maximize the volume, we need to maximize x. The perimeter of the cardboard is 60 inches, so the perimeter of the square that is cut from each corner is 60 - 4x inches. The area of the square that is cut from each corner is (60 - 4x)^2 square inches. The area of the cardboard that is left after the squares are cut out is 60^2 - 4(60 - 4x)^2 square inches. The volume of the box is V = x^3 = (60^2 - 4(60 - 4x)^2)^3/27 cubic inches. To maximize the volume, we need to find the value of x that maximizes the expression (60^2 - 4(60 - 4x)^2)^3/27. We can do this by taking the derivative of the expression with respect to x and setting it equal to 0. The derivative is dV/dx = 4x(60^2 - 4(60 - 4x)^2)^2/27. Setting this equal to 0, we get 4x(60^2 - 4(60 - 4x)^2)^2/27 = 0. This implies that either x = 0 or (60^2 - 4(60 - 4x)^2)^2 = 0. If x = 0, then the volume of the box is 0. If (60^2 - 4(60 - 4x)^2)^2 = 0, then 60^2 - 4(60 - 4x)^2 = 0. Solving for x, we get x = 15. Therefore, the maximum volume of the box is V = (60^2 - 4(60 - 4(15))^2)^3/27 = 16,000 cubic inches.

Question 5

A water line runs east-west. A town wants to connect two new housing developments to the line by running lines from a single point on the existing line to the two developments. The first development is 5 miles south of the existing line; the other development is 7 miles south of the existing line and 3 miles east of the first development. How many miles east from the point due north of the first development should the connection be made to minimize the total length of new line?

Accepted Answer

The answer of A water line runs east-west. A town...

Question 6

Determine the equation of the slant asymptote of  .

Accepted Answer

To find the slant asymptote of a function, we need to divide the polynomial in the numerator by the polynomial in the denominator using long division or synthetic division. The quotient will be the equation of the slant asymptote. 
Performing synthetic division, we get:
           ___________________
(x-8) | 11eaa88b_922e_ea8a_a696_0b4d831536f4_TB5869_11
          -11eaa88b_922f_38ac_a696_d7e6476f72b7_TB5869_88
          ___________________
                     88a2_ea8a_a696_68470b0c6404

Therefore, the equation of the slant asymptote is 88a2_ea8a_a696_68470b0c6404_TB5869. 
So, the best choice is B.

Question 7

Suppose a snowball melts in such a way that it maintains a spherical shape. If the radius is decreasing at a rate of 0.75 cm per hour when the radius is 10 cm, how fast is the volume of the snowball decreasing at that instant?

Accepted Answer

The volume of a sphere is given by $V = \frac{4}{3}\pi r^3$. Differentiating both sides with respect to time ($t$) gives $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$. Substituting $r = 10$ cm and $\frac{dr}{dt} = -0.75$ cm/hr (negative because the radius is decreasing), we get $\frac{dV}{dt} = 4\pi (10)^2 (-0.75) = -300\pi$ cm$^3$/hr. The negative sign indicates the volume is decreasing. The numerical value of $-300\pi$ is approximately $-942.5$ cm$^3$/hr, hence the correct answer is 942.5 cm$^3$/hr.

Question 8

Find the point on the curve  closest to the point  . (Approximate the coordinates of the point to 3 decimal places.)

Accepted Answer

The answer of Find the point on the curve ...

Question 9

Find the slant asymptote and graph the function and its asymptote on the same axes.

Accepted Answer

The answer of Find the slant asymptote and graph the...

Question 10

Oil spills out of a tanker at the rate of 145 gallons per minute. The oil spreads in a circle with a thickness of 1/4 inch. Given that 1 ft³ equals 7.5 gallons, determine the rate at which the radius of the spill is increasing when the radius reaches 90 feet.

Accepted Answer

The volume of oil spilled per minute is 145 gallons. Converting gallons to cubic feet using the conversion factor (1 cubic foot = 7.5 gallons), we get the volume of oil spilled per minute in cubic feet as $145 \div 7.5$. The oil forms a cylindrical shape where the volume $V$ can be expressed as $V = \pi r^2h$, where $r$ is the radius and $h$ is the height (or thickness) of the oil spill. Given the thickness $h = \frac{1}{4}$ inch, we convert this to feet ($\frac{1}{4} \div 12$ feet since there are 12 inches in a foot). To find the rate at which the radius of the spill is increasing, we differentiate the volume with respect to time, $dV/dt = 2\pi rh(dr/dt)$, and solve for $dr/dt$, the rate of change of the radius. Substituting the given values and solving for $dr/dt$ gives us the rate at which the radius is increasing when the radius is 90 feet.

Question 11

A handbill is to contain 98 square inches of printed material, with 4-inch margins at the top and bottom and 2-inch margins on each side. What dimensions for the handbill would use the least amount of paper?

Accepted Answer

We need to find the dimensions that minimize the amount of paper used. Let the width of the printed portion be x inches. Then the height of the printed portion would be 98/x inches. Adding the margins, the total width would be x+4 inches and the total height would be 98/x+4 inches. Thus, the total area of the handbill would be:

A(x) = (x+4) * (98/x+4)

To minimize the amount of paper used, we need to find the minimum of A(x). Taking the derivative and setting it to zero, we get:

A'(x) = -98/(x+4)^2 + 1 = 0

Solving for x, we get x=11. Plugging this back into A(x), we get A(11) = 242 square inches, which is the smallest area possible. Therefore, the dimensions that would use the least amount of paper are 11 inches by 22 inches (choice C).

Question 12

Graph  and its slant asymptote on the same set of axes.

Accepted Answer

The given rational function has a vertical asymptote at x=3, a horizontal asymptote at y=0 and a slant asymptote y=x/2. To graph both the slant asymptote and the function, we can divide the numerator polynomial by the denominator using polynomial long division. The result will be the slant asymptote plus a remainder whose degree is less than the degree of the denominator. The quotient is y=(x-3)/2, so the slant asymptote is y=x/2 - 3/2. The remainder is (-5x+13)/(x-3), which can be graphed separately. Therefore, option B is the correct choice.

Question 13

Find a function whose graph has the given asymptotes.   and

Accepted Answer

The answer of Find a function whose graph has the...

Question 14

Bubba, a farmer, wants to build a pen using the 20 ft  40 ft corner of his barn as part of the pen as shown in the figure below (the corner will not require any fencing). If Bubba has 124 ft of available fencing, then what will be the maximum area?

Accepted Answer

To maximize the area, we need to create a rectangle with sides parallel to the sides of the barn.
Let the length of the pen be x and the width be y. Then we have:
2x + y = 124 - 20 - 40 = 64
y = 64 - 2x
We want to maximize the area A = xy. Substituting for y, we get:
A = x(64 - 2x) = 64x - 2x^2
To find the maximum area, we can find the vertex of the parabola y = -2x^2 + 64x. The x-coordinate of the vertex is -b/2a = -64/(2(-2)) = 16. Substituting x = 16, we get y = 32. Therefore, the maximum area is A = 16 * 32 = 2116 square feet.

Question 15

Determine the following significant features of  , approximating if necessary, and sketch the graph.
a). intercepts
b). asymptotes
c). extrema
d). inflection points

Accepted Answer

The answer of Determine the following significant features of ...

Question 16

Find the point on the curve  closest to the point  .

Accepted Answer

To find the point on the curve closest to another point, we need to calculate the distance between the two points and find the point on the curve that is perpendicular to the line connecting the two points. This is the point of intersection between the curve and the perpendicular line. Using this method, we can calculate that the point on the curve closest to the given point is (2, -2) in the coordinate plane. Converting this back to hexadecimal format, we get the point 11eaa88b_9232_4608_a696_edeb721cadde_TB5869_11. Therefore, the correct answer is choice D.

Question 17

Determine the following significant features of  , approximating if necessary, and sketch the graph.
a). intercepts b). asymptotes
c). extrema d). inflection points

Accepted Answer

The answer of Determine the following significant features of ...

Question 18

Bubba, a farmer, wants to fence-in a three-sided area next to his barn, using the side of the barn as the fourth side of the rectangle. He also wants to divide this rectangular area into three identical pens (see figure below). If Bubba has 104 feet of available fencing, then each pen will have what dimensions in order to maximize its area? Round dimensions to nearest tenth.

Accepted Answer

Let x be the length of one of the sides of the three pens. Then the perimeter of one pen is x+x+2x=4x. The perimeter of all three pens is 12x. Since Bubba has only 104 feet of fencing available, this means that 12x=104 or x=26.

Since one side of each pen is the barn, the rectangular area of each pen is 26*(104-2x)/3. Plugging in x=26, we get each pen has an area of 52*13=676 square feet.

To maximize the area, we want the pens to be as square as possible. Therefore, each pen has dimensions of 26 feet by 13 feet.

Question 19

A ladder 36 feet long is leaning against a vertical wall. If the bottom of the ladder is pulled away from the base of the building at a rate of 3 ft/sec, how fast is the top of the ladder sliding down the wall when it is 28 feet above the ground? Round answer to nearest tenth.

Accepted Answer

Using the Pythagorean theorem, the distance of the bottom of the ladder from the wall is sqrt(36^2 - 28^2) = sqrt(1296 - 784) = sqrt(512) = 22.6 feet. Differentiating both sides of the Pythagorean theorem with respect to time gives 2x(dx/dt) + 2y(dy/dt) = 0, where x is the distance from the wall, y is the height of the ladder on the wall, dx/dt = 3 ft/sec (rate at which the bottom is moving away from the wall), and dy/dt is what we're solving for. Plugging in the values gives 2(22.6)(3) + 2(28)(dy/dt) = 0. Solving for dy/dt gives dy/dt = -2.4 ft/sec, indicating the top of the ladder is sliding down at 2.4 ft/sec.

Question 20

A rectangle has its two bottom corners lying on the x-axis and the two top corners on the parabola  , where  . If the area of the rectangle is to be maximized, then what are its dimensions?

Accepted Answer

The answer of A rectangle has its two bottom corners...

Quiz 4: Applications of the Derivative

A Norman window has the outline of a semicircle on top of a rectangle. Suppose there is feet of wood trim available. Find the dimensions of the rectangle (and, hence, the semicircle) that will maximize the area of the window.

A company's revenue for selling x (thousand) items is given by in millions of dollars. Find the value of x (rounded to the nearest item) that maximizes the revenue and find the maximum revenue (rounded to the nearest dollar) .

Find the volume of the largest box with no top that can be made from a piece of cardboard 60 inches square by cutting equal squares from the corners and turning up the sides.

Determine the equation of the slant asymptote of .

Suppose a snowball melts in such a way that it maintains a spherical shape. If the radius is decreasing at a rate of 0.75 cm per hour when the radius is 10 cm, how fast is the volume of the snowball decreasing at that instant?

Find the point on the curve closest to the point . (Approximate the coordinates of the point to 3 decimal places.)

Find the slant asymptote and graph the function and its asymptote on the same axes.

Oil spills out of a tanker at the rate of 145 gallons per minute. The oil spreads in a circle with a thickness of 1/4 inch. Given that 1 ft³ equals 7.5 gallons, determine the rate at which the radius of the spill is increasing when the radius reaches 90 feet.

A handbill is to contain 98 square inches of printed material, with 4-inch margins at the top and bottom and 2-inch margins on each side. What dimensions for the handbill would use the least amount of paper?

Graph and its slant asymptote on the same set of axes.

Find a function whose graph has the given asymptotes. and

Bubba, a farmer, wants to build a pen using the 20 ft 40 ft corner of his barn as part of the pen as shown in the figure below (the corner will not require any fencing) . If Bubba has 124 ft of available fencing, then what will be the maximum area?

Determine the following significant features of , approximating if necessary, and sketch the graph. a). intercepts b). asymptotes c). extrema d). inflection points

Find the point on the curve closest to the point .

Determine the following significant features of , approximating if necessary, and sketch the graph. a). intercepts b). asymptotes c). extrema d). inflection points

A ladder 36 feet long is leaning against a vertical wall. If the bottom of the ladder is pulled away from the base of the building at a rate of 3 ft/sec, how fast is the top of the ladder sliding down the wall when it is 28 feet above the ground? Round answer to nearest tenth.

A rectangle has its two bottom corners lying on the x-axis and the two top corners on the parabola , where . If the area of the rectangle is to be maximized, then what are its dimensions?

Quiz 4: Applications of the Derivative

A Norman window has the outline of a semicircle on top of a rectangle. Suppose there is feet of wood trim available. Find the dimensions of the rectangle (and, hence, the semicircle) that will maximize the area of the window.

A company's revenue for selling x (thousand) items is given by in millions of dollars. Find the value of x (rounded to the nearest item) that maximizes the revenue and find the maximum revenue (rounded to the nearest dollar) .

Find the volume of the largest box with no top that can be made from a piece of cardboard 60 inches square by cutting equal squares from the corners and turning up the sides.

Determine the equation of the slant asymptote of .

Suppose a snowball melts in such a way that it maintains a spherical shape. If the radius is decreasing at a rate of 0.75 cm per hour when the radius is 10 cm, how fast is the volume of the snowball decreasing at that instant?

Find the point on the curve closest to the point . (Approximate the coordinates of the point to 3 decimal places.)

Find the slant asymptote and graph the function and its asymptote on the same axes.

Oil spills out of a tanker at the rate of 145 gallons per minute. The oil spreads in a circle with a thickness of 1/4 inch. Given that 1 ft3 equals 7.5 gallons, determine the rate at which the radius of the spill is increasing when the radius reaches 90 feet.

A handbill is to contain 98 square inches of printed material, with 4-inch margins at the top and bottom and 2-inch margins on each side. What dimensions for the handbill would use the least amount of paper?

Graph and its slant asymptote on the same set of axes.

Find a function whose graph has the given asymptotes. and

Bubba, a farmer, wants to build a pen using the 20 ft 40 ft corner of his barn as part of the pen as shown in the figure below (the corner will not require any fencing) . If Bubba has 124 ft of available fencing, then what will be the maximum area?

Determine the following significant features of , approximating if necessary, and sketch the graph. a). intercepts b). asymptotes c). extrema d). inflection points

Find the point on the curve closest to the point .

Determine the following significant features of , approximating if necessary, and sketch the graph. a). intercepts b). asymptotes c). extrema d). inflection points

A ladder 36 feet long is leaning against a vertical wall. If the bottom of the ladder is pulled away from the base of the building at a rate of 3 ft/sec, how fast is the top of the ladder sliding down the wall when it is 28 feet above the ground? Round answer to nearest tenth.

A rectangle has its two bottom corners lying on the x-axis and the two top corners on the parabola , where . If the area of the rectangle is to be maximized, then what are its dimensions?

Oil spills out of a tanker at the rate of 145 gallons per minute. The oil spreads in a circle with a thickness of 1/4 inch. Given that 1 ft³ equals 7.5 gallons, determine the rate at which the radius of the spill is increasing when the radius reaches 90 feet.