Question 1

A gas pipeline is to be constructed from a storage tank, which is right on a road, to a house which is 600 feet down the road and 300 feet back from the road. Pipe laid along the road costs $8.00 per foot, while pipe laid o the road costs $10.00 per foot. What is the minimum cost for which this pipeline can be built?

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Question 2

A hog weighs 250 pounds. A high-yield diet allows the animal to gain 6 pounds a day at a cost of $0.56 a day. The market price for hogs is currently $0.75 per pound, but it is falling at a rate of $0.01 per day, and that price decline is expected to remain steady for the foreseeable future. When should the hog be sold in order to provide the farmer with the highest financial gain?

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Question 3

An open box is made from a 16 inch $\times$ 16 inch piece of cardboard by cutting equal squares from each corner and folding up the sides. For maximum volume, what size squares should be cut out?

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Question 4

A rectangle with base on the x-axis has its upper vertices on the curve $y = 4 - \frac { 1 } { 2 } x ^ { 2 }$ . Find the maximum perimeter of such a rectangle.

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Question 5

A company estimates that the initial cost of producing a commodity would be $680: The additional cost of manufacturing each item can be modeled by the function $m ( x ) = 3 x + 0.01 x ^ { 2 }$ , where x is the number of items produced and m is the manufacturing cost, in dollars. A market survey indicates the demand function can be estimated by $p ( x ) = 12 - 0.002 x$ .(a) Find the production level that maximizes profit. What is the maximum profit?
(b) What should the selling price be to maximize the profit?

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Question 6

What is the smallest slope for a tangent line to the graph of $y = 2 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 1$ ?

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Question 7

A large, closed shipping container with a square base is to be made from 1000 square feet of fiberboard. Find the dimensions of the container with the greatest volume.

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Question 8

A 400-meter-long running track is to be constructed with two parallel straightaways of length s joining two semi-circular curves of radius r. What are the dimensions of the track that will enclose the largest infield area?

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Question 9

A rectangle with base on the x-axis has its upper vertices on the curve $y = 12 - x ^ { 2 }$ . Find the maximum area of such a rectangle.

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Question 10

A drinking cup is made in the shape of a right circular cylinder. For a fixed volume, we wish to make the total material used (the circular bottom and the cylindrical side) as small as possible. Under this condition, what is the ratio of the height to the diameter?

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Question 11

A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area can be enclosed?

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Question 12

A rancher intends to fence o a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed?

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Question 13

A lakefront runs east-west. A man in a rowboat is 5 miles due north of point A on the shore. He wishes to get to B, 5 miles due east of A, in the least time. He is able to row 3 miles per hour and walk 5 miles per hour. What is the minimum time, in minutes?

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Question 14

A rental estate management company manages an apartment complex with 20 units. The manager estimates that all 20 units can be rented if the rent is $150 per unit per month, and that for each increase in rent of $25, one apartment will be vacated. What rent should the manager charge to maximize revenue?

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Question 15

It is desired to make a rectangular pen with diameter equal to 40 feet. It is not necessary, however, to achieve maximum area. In fact, any area that is greater than or equal to 75% of the maximum area will be satisfactory. Under these conditions, find the maximum acceptable length of the longer side.

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Question 16

A rectangular region with area 3200 square feet is to be enclosed within a fence. The two sides which run north-south will use fencing materials costing $1.00 per foot, while the other two sides require fencing materials which cost $2.00 per foot. Find the dimensions of the region which gives the smallest material cost.

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Question 17

A poster is to have a total area of 486 square inches with 3-inch margins at the bottom and top and a 2-inch margin at each side. What dimensions of a poster will give the largest printed area?

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Question 18

Amy is 2 miles offshore in a boat and wishes to reach a coastal village 6 miles down a straight shoreline from the point nearest to the boat. She can row 2 miles per hour and can walk 5 miles per hour. Where should she land her boat to reach the village in the least amount of time? How long will it take Amy to reach the village?

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Question 19

A rectangular storage container with an open top is to have a volume of 10 m $3$ : The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the dimensions of the container needed to minimize the cost.

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Question 20

A cylindrical can is to be made with volume $V _ { 0 }$ . Find dimensions that will minimize the amount of material used in its construction. (Assume the thickness of the material is uniform.)

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Quiz 4: Applications of Differentiation

An open box is made from a 16 inch ×\times× 16 inch piece of cardboard by cutting equal squares from each corner and folding up the sides. For maximum volume, what size squares should be cut out?

A rectangle with base on the x-axis has its upper vertices on the curve y=4−12x2y = 4 - \frac { 1 } { 2 } x ^ { 2 }y=4−21​x2 . Find the maximum perimeter of such a rectangle.

What is the smallest slope for a tangent line to the graph of y=2x3−2x2+5x+1y = 2 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 1y=2x3−2x2+5x+1 ?