Section 3.5 Extra Exercises

1) A rectangular box with a volume of 500 cu ft to be constructed with square base and top. The cost per square feet for the base is $2, for the top is $3 and for the sides is $5. What dimensions will minimize the cost? what is the cost?

(Ans: 10, 10, 5, 1500)

2) A refrigerator wholesaler expects to sell 2500 refrigerators per year. It costs $10 to store one refrigerator for one year, to reorder there is a fixed cost of $20 plus $9 for each refrigerator ordered. How many times per year and in what lot size should the store reorder to minimize the inventory cost?

(Ans: 100, 25)

3) A rectangular lot to be fenced off along the highway, the fence along the highway costs $4 per ft and on the other sides $2 per ft. Find the area of the largest lot that can be fenced off for $600.

(Ans: 3750)

4) From a piece of a cardboard 100 in by 100 in, square corners are cut so the sides can be folded up make a box. What dimensions will yield a box of maximum volume?

(Ans: 66.67, 16.67)

5) A rectangular lot to be fenced off along the highway, the fence along the highway costs $5 per ft, on the two sides is $3 per foot and on the middle is side is $2 per ft. Find the dimensions for the largest area that can be fenced off for $800.

(Ans: 80, 50)

6) A farmer wants to enclose three rectangular areas next to a river using 300 feet of fencing. What is the largest area that can be enclosed? (Note that the farmer doesn't have to fence the sides next to the river and the road)

(Ans: 7500)

7) A theater owner charges $4 for admission when there is an average attendance of 100 people. for every $0.20 increase in admission, there is a loss of 2 customers. What admission should be charged to maximize the revenue? what is the number of people that will maximize the revenue?

(Ans: 7, 70)

8) A rectangular play area to be fenced off beside Bob's house. Bob agreed to pay the cost of the side beside his house, also he agreed to pay 1/2 of the cost for two sides and 1/3 of the cost for the other side that he shares with his neighbors. if he has $480, what are the dimensions for the largest area?

(Ans: 180, 240)

9) A gardener wishes to fence in a rectangular area of 1728 square feet. She also wants to insert a fence that will divide the area into two rectangular sub-areas. The drawing shows that some fencing costs $4 per foot and some costs $2 per foot. Find the dimensions that will minimize the cost of the garden fencing.

(Ans: 48 feet vertical by 36 feet horizontal)

10) An oil company wants to lay a pipeline from its offshore drilling rig to a storage tank on shore. The rig (point R) is 3 miles offshore (point A), and the storage tank (point B) is 8 miles down the shoreline. The costs of laying pipe underwater are $800 per mile and along the shoreline are $400 per mile. Point P is the point on shore where the underwater pipe connects with the shoreline pipe. Where should point P be located so as to minimize the cost?

(Ans: x = 1.732, distance BP = 6.268 miles)

11) A theater determine that if the admission price is $ 30, it averages 400 people in attendance. But for every increase of $ 2, it loses 20 customers from the average number. Every customers spends an average of $ 2 on concessions. What admission price should the theater charge in order to maximize the revenue?

(Ans: $34)

12) A driver is trying to reach the hospital from an accident site (Point S) which was located 12 miles from the paved road (Point A). The nearest hospital is located 11 miles down the paved road (Point B). If he can drive an average speed of 20 mph on the desert and 52 mph on the road, locate the point P on the road toward which he should drive in order to minimize the time needed to get to the hospital. (hint: Time=distance/speed)

(Ans: x = 5 miles from point A)

13) A company estimate that it can sell 1000 units per week if it sets the unit price at $5, but that its weekly sales will rise by 100 units for each $0.10 decrease in price. The company has a fixed cost each week of $1050 and the costs in labor and materials to make a unit is $0.10. Find the production level that maximize the profit.

(Ans: 2950 units)

14) An apartment complex has 80 units. When the rent is $400 per month, all units are rented. For each $10 increase in rent, one apartment unit becomes vacant. What rent should be charged to produce the maximum revenue?

(Ans: $600)

15) John estimated that his farm would yield 1000 bushels of sugar beets if harvested right now. Today's beet price is $2.5 per bushel, but it will drop $0.05 a bushel each day from now on. On the other hand, John guesses that the crop is still growing at 30 bushels per day. In how many days from now should he harvest and sell to maximize the revenue?

(Ans: 8.3 or 8 days)

16) An apartment complex with 220 units can fill all units when the rent is $480 per month. It is estimated that for each $15 per month increase in rent, 5 units will become vacant. The complex has a fixed monthly costs of $60,000 and a maintenance cost of $60 for each unit rented. What monthly rent should be charged to maximize the profit?

(Ans: $600)

17) An apple grower has 600 bushels of apples which he can sell to a single wholesaler. Today he could get $10 per bushel, and the price is going up $0.25 per day. On the other hand, he can count on spoilage of about 8 bushels per day. When should he sell the apples in order to maximize the revenue?

(Ans: between 17 to 18 days)

18) A company manufactures a machines has a fixed monthly cost of $1000 and direct costs of $8 for each machine produced. The company estimates that 100 machines can be sold if the unit price is $40, and that 10 more machines will be sold for each decrease of $2 in the price. Find the price per unit and the number of units that will maximize the profit.

(Ans: $34 per unit, 130 units)

19) Mark likes to use 100 feet of fencing to fence a rectangular garden next to his house. What dimensions will give the maximum garden area? (see the graph, note that there is no fence next to the house)

(Ans: x= 31 feet, y= 31 feet)

20) The U.S. Postal Service has the following restriction on mailing a fourth-class parcel in the form of a rectangular box. The perimeter of one end plus the length of the box must be no more than 108 inches. What is the largest volume of a permissible rectangular parcel whose ends are square?

(Ans: 11664 cubic inches)