Optimization

Min-Max Problems

Problem 1

A car rental agency rents 200 cars per day at a rate of $30 per day. For each $1 increase in rate, 5 fewer cars are rented. At what rate should the cars be rented to produce the maximum income? What is the maximum income?

1. R(x) = (200 – 5x)(30 + x)R(x) = 6000 + 50x – 5x2

R ’ (x) = 50 – 10xR ‘ (x) = 0 50 – 10x = 0 x = 5

R “ (x) = –10 so f ” (x) is concave down everywhere, so x = 5 is an amax

$35/day or 175 cars/day; $6125

Answer to Problem 1

Problem 2

A candy box is to be made out of a piece of cardboard that measures 8 by 12 inches. Squares of equal size will be cut out of each corner, and then the ends andsides will be folded up to form a rectangular box. What size square should be cut from each corner to obtain a maximum volume?

Answer to Problem 2

V(x) = x(8 - 2x)(12 - 2x); 0 < x < 4V(x) = 96x – 40x2 + 4x3

V ‘ (x) = 96 – 80x + 12x2

V ‘ (x) = 0 12x2 – 80x + 96 = 03x2 – 20x + 24 = 0

x = 5.097 in. or 1.569 in., but 0 < x < 4therefore x = approx. 1.57 in

Problem 3A family plans to fence in a rectangular patio area behind their house. They have 120 feet of fence to use. One side of the rectangle is the back of the house. What should be the dimensions of the rectangular region if they want to make the enclosed patio area as large as possible.

Answer to Problem 3

A(x) = x(120 – 2x) A(x) = 120x – 2x2

A ‘ (x) = 120 – 4xA ‘ (x) = 0 120 – 4x = 0 x = 30; 30ft x 60ft = 1800 ft2

Problem 4A manufacturer of storage bins plans to produce some open-top rectangular boxes with square bases. The volume of each box is to be 125 cubic feet. material forthe base costs $6 per square foot, and material for the sides costs $3 per square foot. Determine the dimensions of the box that will minimize the cost of materials.

Answer to Problem 4

5ft X 5ft X 5ft

€

C x( ) = $6x 2 + $3⋅4x125

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟

C x⎛⎝⎜

⎞

⎠⎟=6x2+1500x =6x2+1500x−1

C ‘ (x) = 12x – 1500x–2

C ‘ (x) = 0 12x – 1500x–2 = 0

12x = 1500x–2 12x3 = 1500x3 = 125 x = 5

Problem 5The owner of a warehouse decides to fence in an area of 800 square feet behind the warehouse. He plans to use the wall of the building as one of the four sides that will enclose the rectangular area. He would like to use the least amount of fencing necessary for the other three sides. How many feet of fence will be needed.

Answer to Problem 5

€

f x( ) = 2x +800

x;20 ft × 40 ft

80 ft. of fence

Problem 6

Given an isosceles triangle with equal sides of 3 inches in length. Determine the measure of the angle between the two equal sides which results in the largest area.

Answer to Problem 6

90 degrees

Problem 7Construct a window in the shape of a semi-circle over a rectangle (like many stained glass windows in churches). If the distance around the outside of the window is 12 feet, what dimensions will result in the rectangle having the largest possible area?

Answer to Problem 7

Answers: The rectangle's dimensions would be 2.33 feet by 3 feet, so the area would be 6.99 square feet.

P = 2r + πr + 2y

12 = 2r + πr + 2y 12 – 2r – πr = 2y

(12 – 2r – πr)/2 = y

A = 2ry

A = 2r (12 – 2r – πr)/2

A = r (12 – 2r – πr)

A = 12r – 2r2 – πr2

A = – 2r2 – πr2 + 12r

A’ = – 4r – 2πr + 12

0 = – 4r – 2πr + 12

4r + 2πr = 12 r = 12/(4 + 2π) = 1.66

Problem 8Ladder in the Hall Problem A non-folding ladder is to be taken around a corner where 2 hallways intersect at right angles. One hall is 7' wide and the other hall is 5' wide. What is the maximum length that the ladder can be so it will pass around the corner?

Answer to Problem 8

Answer: 16.89 feet

Problem 9During the Winter break, Jeremy Davis is enroute to his uncle's house. Two policemen are stationed two miles apart along the road, which has a posted speed limit of 55 mph. The first clocks him at 50 mph as he passes, and the second policeman clocks him at 55 mph, butpulls him over anyway. When he asks why he got pulled over, the policeman responds, "It took you 90 seconds to travel 2 miles." Why is he guilty of speeding according to the Mean Value Theorem?

Answer to Problem 9

Answer: His average rate is 80 mph. There must have been at least one point over the interval where the instantaneous rate of change (velocity) must equal the average rate of change (average speed). Therefore, old lead foot Taylor must have traveled 80 mph at least once, and the policeman can ticket him.

Problem 10The practice of shooting bullets into the air for whatever purpose is extremely dangerous. Assuming that a hunting rifle discharges a bullet with an initial velocity of 3000 ft/sec from a height of 6 feet, answer the following questions: (A) How high will the bullet travel at its peak? (B) At what speed will the bullet be traveling when it slams into the ground, assuming that it hits nothing in its path?

Answer to Problem 10

Answers: (A) 140,631 feet (it takes 93.75 seconds to reach the maximum height) (B) 3000.064 ft/sec

Problem 11

To celebrate our 25th Anniversary in 2008, I commissioned the construction of a four-inch tall box made of precious metals to give to my wife. The jewelry box will have rectangular sides and an open top. The longer sides of the box were to be made of gold, at a cost of $300 per in2; the shorter sides were to be made of platinum, at a cost of $550 per in2. The bottom as to be made of plywood, at a cost of 2 cents per in2.

What dimensions provide me with the lowest cost if I am adamant that the box have a volume of 50 cubic inches?

Answer to Problem 11

Answer: C = 2400x + 4400y + xy(.02)

Dimensions are 4.787 in x 2.611 in x 4 in.

(The lowest cost for the box will be $23,000, do I still need to buy a card?)

Problem 12

A builder is purchasing a rectangular plot of land with frontage on a road for the purpose of constructing a rectangular warehouse. Its floor area must be 300,000 square feet. Local building codes require that the building be set back 40 feet from the road and that there be empty buffer strips of land 25 feet wide on the sides and 20 feet in the back. Find the overall dimensions of the parcel of land and building which will minimize the area of the land parcel that the builder must buy.

Answer to Problem 12

Answer: Dimensions of the building: 500' x 600' Dimensions of the land: 550' x 660'