Question 1

The cost of fuel used in propelling a dirigible varies as the square of its speed and is $100/hour when the speed is 100 miles/hour. Other expenses amount to $200/hour. Find the most economical speed for a voyage of 1,400 miles.

Accepted Answer

The answer of The cost of fuel used in propelling...

Question 2

A rancher is going to build a 3-sided cattle enclosure with a divider down the middle as shown to the right. The cost per foot of the 3 side walls will be $4/foot, while the back wall, being taller, will be $8/foot. If the rancher wishes to enclose an area of 384 ft², what dimensions of the enclosure will minimize his cost?

Accepted Answer

The answer of A rancher is going to build a...

Question 3

The stiffness of a beam of rectangular cross section is proportional to the product xy³. Find the stiffest beam which can be cut from a log with diameter d = 14 inches. See figure on right.

Accepted Answer

The answer of The stiffness of a beam of rectangular...

Question 4

A rectangle is to have an area of 32 in². What should be its dimensions if the distance from one corner to the mid-point of a nonadjacent side is to be a minimum?

Accepted Answer

The answer of A rectangle is to have an area...

Question 5

Find values for x and y such that their product is a minimum if y = 7x -8.

Accepted Answer

The answer of Find values for x and y such...

Question 6

An open cylindrical trash can is to hold 11 cubic feet of material. What should be its dimensions if the cost of material used is to be a minimum? [Surface Area, S = $\pi$r² + 2 $\pi$rh where r = radius and h = height.]

Accepted Answer

The answer of An open cylindrical trash can is to...

Question 7

Find the circular cylinder of largest lateral area which can be inscribed in a sphere of radius R = 4 feet. [Surface area of a cylinder, S = 2 $\pi$rh, where r = radius, h = height.]

Accepted Answer

The answer of Find the circular cylinder of largest lateral...

Question 8

An open-top shipping crate with square bottom and rectangular sides is to hold 108 in³ and requires a minimum amount of cardboard. Find the most economical dimensions.

Accepted Answer

The answer of An open-top shipping crate with square bottom...

Question 9

The graph below depicts the position function of a particle moving on a coordinate line at three different times. For each time specify whether the particle is moving to the left or right (Assume right is the positive direction.) and whether or not it is speeding up or slowing.

Accepted Answer

The answer of The graph below depicts the position function...

Question 10

A rectangular garden is to be laid out with one side adjoining a neighbor's lot and is to contain 507ft². If the neighbor agrees to pay for half the dividing fence, what should the dimensions of the garden be to insure a minimum cost of enclosure?

Accepted Answer

The answer of A rectangular garden is to be laid...

Question 11

A can containing 16 in³ of tuna and water is to be made in the form of a circular cylinder. What dimensions of the can will require the least amount of material? (V = $\pi$r²h, S = 2 $\pi$rh, A = $\pi$r²)

Accepted Answer

The answer of A can containing 16 in³ of tuna...

Question 12

The product of 2 positive numbers is 1,875. Find the numbers, if the sum of one number and the cube of the other is to be minimum.

Accepted Answer

The answer of The product of 2 positive numbers is...

Question 13

A poster is to contain 240 in² of printed matter with margins of 3 in. each at top and bottom and 2 in. at each side. Find the overall dimensions if the total area of the poster is to be a minimum.

Accepted Answer

The answer of A poster is to contain 240 in²...

Question 14

Find the point on the parabola 2y = x² which is closest to (12, 17).

Accepted Answer

The answer of Find the point on the parabola 2y...

Question 15

A line is drawn through the point P(5, 2) so that it intersects the y-axis at A(0,y) and the x-axis at B(x,0). Find the smallest triangle formed if x and y are positive.

Accepted Answer

The answer of A line is drawn through the point...

Question 16

A slice of pizza, in the form of a sector of a circle, is to have a perimeter of 18 inches. What should be the radius of the pan to make the slice of pizza largest.
(The area of a sector of a circle,  where $\theta$ is the central angle in radians and the arc length along a circle is S = r$\theta$with $\theta$ in radians.)

Accepted Answer

The answer of A slice of pizza, in the form...

Question 17

Two fences, a distance of d = 16 feet apart are to be constructed so that the first fence is a height of h = 2 feet high and the second fence is higher than the first. What is the length of the shortest pole that has one end on the ground, passing over the first fence and reaches the second fence? See figure.

Accepted Answer

The answer of Two fences, a distance of d =...

Question 18

Find the dimensions of the rectangular area of maximum area which can be laid out within a triangle of base 16 and altitude 4 if one side of the rectangle lies on the base of the triangle.

Accepted Answer

The answer of Find the dimensions of the rectangular area...

Question 19

Find the maximum sum of 2 numbers given that the first plus the square of the second is equal to 70?

Accepted Answer

The answer of Find the maximum sum of 2 numbers...

Question 20

If three sides of a trapezoid are a = 8 inches long, how long should the fourth side be if the area is a maximum? [Area of a trapezoid =  where a and b are the lengths of the parallel sides and h = height.]

Accepted Answer

The answer of If three sides of a trapezoid are...

Quiz 4: The Derivative in Graphing and Applications

The cost of fuel used in propelling a dirigible varies as the square of its speed and is $100/hour when the speed is 100 miles/hour. Other expenses amount to $200/hour. Find the most economical speed for a voyage of 1,400 miles.

The stiffness of a beam of rectangular cross section is proportional to the product xy³. Find the stiffest beam which can be cut from a log with diameter d = 14 inches. See figure on right.

A rectangle is to have an area of 32 in². What should be its dimensions if the distance from one corner to the mid-point of a nonadjacent side is to be a minimum?

Find values for x and y such that their product is a minimum if y = 7x -8.

An open cylindrical trash can is to hold 11 cubic feet of material. What should be its dimensions if the cost of material used is to be a minimum? [Surface Area, S = $\pi$ r² + 2 $\pi$ rh where r = radius and h = height.]

Find the circular cylinder of largest lateral area which can be inscribed in a sphere of radius R = 4 feet. [Surface area of a cylinder, S = 2 $\pi$ rh, where r = radius, h = height.]

An open-top shipping crate with square bottom and rectangular sides is to hold 108 in³ and requires a minimum amount of cardboard. Find the most economical dimensions.

The graph below depicts the position function of a particle moving on a coordinate line at three different times. For each time specify whether the particle is moving to the left or right (Assume right is the positive direction.) and whether or not it is speeding up or slowing.

A rectangular garden is to be laid out with one side adjoining a neighbor's lot and is to contain 507ft². If the neighbor agrees to pay for half the dividing fence, what should the dimensions of the garden be to insure a minimum cost of enclosure?

A can containing 16 in³ of tuna and water is to be made in the form of a circular cylinder. What dimensions of the can will require the least amount of material? (V = $\pi$ r²h, S = 2 $\pi$ rh, A = $\pi$ r²)

The product of 2 positive numbers is 1,875. Find the numbers, if the sum of one number and the cube of the other is to be minimum.

A poster is to contain 240 in² of printed matter with margins of 3 in. each at top and bottom and 2 in. at each side. Find the overall dimensions if the total area of the poster is to be a minimum.

Find the point on the parabola 2y = x² which is closest to (12, 17).

A line is drawn through the point P(5, 2) so that it intersects the y-axis at A(0,y) and the x-axis at B(x,0). Find the smallest triangle formed if x and y are positive.

Find the dimensions of the rectangular area of maximum area which can be laid out within a triangle of base 16 and altitude 4 if one side of the rectangle lies on the base of the triangle.

Find the maximum sum of 2 numbers given that the first plus the square of the second is equal to 70?

If three sides of a trapezoid are a = 8 inches long, how long should the fourth side be if the area is a maximum? [Area of a trapezoid = where a and b are the lengths of the parallel sides and h = height.]

Quiz 4: The Derivative in Graphing and Applications

The cost of fuel used in propelling a dirigible varies as the square of its speed and is $100/hour when the speed is 100 miles/hour. Other expenses amount to $200/hour. Find the most economical speed for a voyage of 1,400 miles.

The stiffness of a beam of rectangular cross section is proportional to the product xy3. Find the stiffest beam which can be cut from a log with diameter d = 14 inches. See figure on right.

A rectangle is to have an area of 32 in2. What should be its dimensions if the distance from one corner to the mid-point of a nonadjacent side is to be a minimum?

Find values for x and y such that their product is a minimum if y = 7x -8.

An open cylindrical trash can is to hold 11 cubic feet of material. What should be its dimensions if the cost of material used is to be a minimum? [Surface Area, S = π\piπ r2 + 2 π\piπ rh where r = radius and h = height.]

Find the circular cylinder of largest lateral area which can be inscribed in a sphere of radius R = 4 feet. [Surface area of a cylinder, S = 2 π\piπ rh, where r = radius, h = height.]

An open-top shipping crate with square bottom and rectangular sides is to hold 108 in3 and requires a minimum amount of cardboard. Find the most economical dimensions.

The graph below depicts the position function of a particle moving on a coordinate line at three different times. For each time specify whether the particle is moving to the left or right (Assume right is the positive direction.) and whether or not it is speeding up or slowing.

A rectangular garden is to be laid out with one side adjoining a neighbor's lot and is to contain 507ft2. If the neighbor agrees to pay for half the dividing fence, what should the dimensions of the garden be to insure a minimum cost of enclosure?

A can containing 16 in3 of tuna and water is to be made in the form of a circular cylinder. What dimensions of the can will require the least amount of material? (V = π\piπ r2h, S = 2 π\piπ rh, A = π\piπ r2)

The product of 2 positive numbers is 1,875. Find the numbers, if the sum of one number and the cube of the other is to be minimum.

A poster is to contain 240 in2 of printed matter with margins of 3 in. each at top and bottom and 2 in. at each side. Find the overall dimensions if the total area of the poster is to be a minimum.

Find the point on the parabola 2y = x2 which is closest to (12, 17).

A line is drawn through the point P(5, 2) so that it intersects the y-axis at A(0,y) and the x-axis at B(x,0). Find the smallest triangle formed if x and y are positive.

Find the dimensions of the rectangular area of maximum area which can be laid out within a triangle of base 16 and altitude 4 if one side of the rectangle lies on the base of the triangle.

Find the maximum sum of 2 numbers given that the first plus the square of the second is equal to 70?

If three sides of a trapezoid are a = 8 inches long, how long should the fourth side be if the area is a maximum? [Area of a trapezoid = where a and b are the lengths of the parallel sides and h = height.]

The stiffness of a beam of rectangular cross section is proportional to the product xy³. Find the stiffest beam which can be cut from a log with diameter d = 14 inches. See figure on right.

A rectangle is to have an area of 32 in². What should be its dimensions if the distance from one corner to the mid-point of a nonadjacent side is to be a minimum?

An open cylindrical trash can is to hold 11 cubic feet of material. What should be its dimensions if the cost of material used is to be a minimum? [Surface Area, S = $\pi$ r² + 2 $\pi$ rh where r = radius and h = height.]

Find the circular cylinder of largest lateral area which can be inscribed in a sphere of radius R = 4 feet. [Surface area of a cylinder, S = 2 $\pi$ rh, where r = radius, h = height.]

An open-top shipping crate with square bottom and rectangular sides is to hold 108 in³ and requires a minimum amount of cardboard. Find the most economical dimensions.

A rectangular garden is to be laid out with one side adjoining a neighbor's lot and is to contain 507ft². If the neighbor agrees to pay for half the dividing fence, what should the dimensions of the garden be to insure a minimum cost of enclosure?

A can containing 16 in³ of tuna and water is to be made in the form of a circular cylinder. What dimensions of the can will require the least amount of material? (V = $\pi$ r²h, S = 2 $\pi$ rh, A = $\pi$ r²)

A poster is to contain 240 in² of printed matter with margins of 3 in. each at top and bottom and 2 in. at each side. Find the overall dimensions if the total area of the poster is to be a minimum.

Find the point on the parabola 2y = x² which is closest to (12, 17).