Question 1

A lighthouse is 8 miles off a straight coast and a town is located 18 miles down the seacoast. Supplies are to be moved from the town to the lighthouse on a regular basis and at a minimum time. If the supplies can be moved at the rate of 7 miles/hour on water and 25 miles/hour over land, how far from the town should a dock be constructed for shipment of supplies?

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Question 2

Find the maximum sum of 2 numbers given that the first plus the square of the second is equal to 28?

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Question 3

Find the dimensions of the rectangle of maximum area which may be embedded in a right triangle with base of 5, height of 12, and hypotenuse of 13 feet as shown in the figure.

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Question 4

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 = 28 inches. See figure on right.

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Question 5

Find the point on the curve x² + y² = 36 closest to (10, 0).

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Question 6

Find the area of the largest possible isosceles triangle with 2 sides equal to 12.

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Question 7

The strength of a beam with a rectangular cross section varies directly as x and as the square of y. What are the dimensions of the strongest beam that can be sawed out of a round log whose diameter is d = 18 inches? See figure on right.

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Question 8

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 $8/foot, while the back wall, being taller, will be $16/foot. If the rancher wishes to enclose an area of 15,000 ft², what dimensions of the enclosure will minimize his cost?

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Question 9

A poster is to contain 180 in² of printed matter with margins of 2 in. each at top and bottom and 2 in. at each side. Find the original dimensions of the poster if the total area of the poster with border is to be a minimum.

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Question 10

An isosceles triangle is drawn with its vertex at the origin and its base parallel to the x axis. The vertices of the base are on the curve 6y = 30 - x². Find the area of the largest such triangle.

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Question 11

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.

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Question 12

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.]

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Question 13

A long strip of copper W = 24 inches wide is to be made into a rain gutter by turning up the sides to form a trough with a rectangular cross section. Find the dimensions of the cross section if the carrying capacity of the trough is to be a maximum.

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Question 14

Find the point on the curve x² + y² = 121 closest to (10, 0).

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Question 15

The infield of a 440 yard track consists of a rectangle and 2 semicircles. To what dimensions should the track be built in order to maximize the area of the rectangle?

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Question 16

A divided field is to be constructed with 2720 feet of fence as shown.
For what value of x will the area be a maximum?

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Question 17

If three sides of a trapezoid are a = 6 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.]

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Question 18

A can containing 13.5 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)

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Question 19

Find the dimensions of the rectangular area of maximum area which can be laid out within a triangle of base 24 and altitude 6 if one side of the rectangle lies on the base of the triangle.

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Question 20

Find the dimensions of the rectangle of greatest area that can be inscribed in a circle of radius a.

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Quiz 4: The Derivative in Graphing and Applications

Find the maximum sum of 2 numbers given that the first plus the square of the second is equal to 28?

Find the dimensions of the rectangle of maximum area which may be embedded in a right triangle with base of 5, height of 12, and hypotenuse of 13 feet as shown in the figure.

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 = 28 inches. See figure on right.

Find the point on the curve x² + y² = 36 closest to (10, 0).

Find the area of the largest possible isosceles triangle with 2 sides equal to 12.

The strength of a beam with a rectangular cross section varies directly as x and as the square of y. What are the dimensions of the strongest beam that can be sawed out of a round log whose diameter is d = 18 inches? See figure on right.

A poster is to contain 180 in² of printed matter with margins of 2 in. each at top and bottom and 2 in. at each side. Find the original dimensions of the poster if the total area of the poster with border is to be a minimum.

An isosceles triangle is drawn with its vertex at the origin and its base parallel to the x axis. The vertices of the base are on the curve 6y = 30 - x². Find the area of the largest such triangle.

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.

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.]

A long strip of copper W = 24 inches wide is to be made into a rain gutter by turning up the sides to form a trough with a rectangular cross section. Find the dimensions of the cross section if the carrying capacity of the trough is to be a maximum.

Find the point on the curve x² + y² = 121 closest to (10, 0).

The infield of a 440 yard track consists of a rectangle and 2 semicircles. To what dimensions should the track be built in order to maximize the area of the rectangle?

A divided field is to be constructed with 2720 feet of fence as shown. For what value of x will the area be a maximum?

If three sides of a trapezoid are a = 6 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.]

A can containing 13.5 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)

Find the dimensions of the rectangular area of maximum area which can be laid out within a triangle of base 24 and altitude 6 if one side of the rectangle lies on the base of the triangle.

Find the dimensions of the rectangle of greatest area that can be inscribed in a circle of radius a.

Quiz 4: The Derivative in Graphing and Applications

Find the maximum sum of 2 numbers given that the first plus the square of the second is equal to 28?

Find the dimensions of the rectangle of maximum area which may be embedded in a right triangle with base of 5, height of 12, and hypotenuse of 13 feet as shown in the figure.

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 = 28 inches. See figure on right.

Find the point on the curve x2 + y2 = 36 closest to (10, 0).

Find the area of the largest possible isosceles triangle with 2 sides equal to 12.

The strength of a beam with a rectangular cross section varies directly as x and as the square of y. What are the dimensions of the strongest beam that can be sawed out of a round log whose diameter is d = 18 inches? See figure on right.

A poster is to contain 180 in2 of printed matter with margins of 2 in. each at top and bottom and 2 in. at each side. Find the original dimensions of the poster if the total area of the poster with border is to be a minimum.

An isosceles triangle is drawn with its vertex at the origin and its base parallel to the x axis. The vertices of the base are on the curve 6y = 30 - x2. Find the area of the largest such triangle.

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.

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.]

A long strip of copper W = 24 inches wide is to be made into a rain gutter by turning up the sides to form a trough with a rectangular cross section. Find the dimensions of the cross section if the carrying capacity of the trough is to be a maximum.

Find the point on the curve x2 + y2 = 121 closest to (10, 0).

The infield of a 440 yard track consists of a rectangle and 2 semicircles. To what dimensions should the track be built in order to maximize the area of the rectangle?

A divided field is to be constructed with 2720 feet of fence as shown. For what value of x will the area be a maximum?

If three sides of a trapezoid are a = 6 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.]

A can containing 13.5 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)

Find the dimensions of the rectangular area of maximum area which can be laid out within a triangle of base 24 and altitude 6 if one side of the rectangle lies on the base of the triangle.

Find the dimensions of the rectangle of greatest area that can be inscribed in a circle of radius a.

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 = 28 inches. See figure on right.

Find the point on the curve x² + y² = 36 closest to (10, 0).

A poster is to contain 180 in² of printed matter with margins of 2 in. each at top and bottom and 2 in. at each side. Find the original dimensions of the poster if the total area of the poster with border is to be a minimum.

An isosceles triangle is drawn with its vertex at the origin and its base parallel to the x axis. The vertices of the base are on the curve 6y = 30 - x². Find the area of the largest such triangle.

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.

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.]

Find the point on the curve x² + y² = 121 closest to (10, 0).

A can containing 13.5 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)