Practice Problems on Optimization - Winona

Practice Problems on Optimization

1. A rectangle is to be inscribed in a semicircle of radius 10. Find the dimensions of the rectangle with the

maximal area.

Answer: W = 5 2 and L = 10 2

A = 2W

100 - W

2

2. Find the dimensions of the rectangle with the largest area that can be inscribed in a right triangle whose

sides are 8 and 12.

Answer: L = 6 and W = 4

A=

2

L (12 - L) (L is the vertical side of

3

the rectangle.)

3

3. A cardboard box with a closed top is to be constructed having a volume of 9 ft and such that the bottom is

a rectangle that is twice as long as it is wide. What should the dimensions of the box be so that it uses the

least amount of cardboard?

Answer: W = 3/2 ft. , L = 3 ft., and H =2 ft.

A = 27/W + 4W

2

4. A rectangular plot of ground is to be fenced on three sides. The fourth side is bounded by an established

hedge. What should the dimensions of the plot be if there are 50 meters of fencing available and if the plot is

to have the largest area?

Answer: W = 12.5 m and L = 25 m

A = (50-2W)W

5. An open top box is to be made out of a 16" by 20" rectangular piece of cardboard by cutting out a square of

side x from each of the four corners and then folding the flaps to form the box. Find the value of x that

maximizes the volume of the resulting box.

Answer: x =

2

{9 3

21}

V = LWH = x(20-2x)(16-2x)

2

2

6. Find the point on the circle x + y = 4 that is closest to the point (5, 12).

Answer: x = 10/13 and y = 24/13.

D=

2

(5 - x) + (12 -

2 2

4-x )

7. Find the shortest distance between the point (3, 4) and the line y = -5x + 2.

Answer:

289/26 .

D=

2

(3 - x) + (4 - (-5x+2))

2

8. Students and faculty from a middle school are renting buses to go on a field trip to a museum. If the

number of passengers is no more than 60, the cost is $20 per passenger. If the number of passengers exceeds

60, then the cost per passenger is reduced by $0.25 for each passenger in excess of 60. If the maximum

number that can join the field trip is 100, what is the number of passengers that maximizes the revenue for the

bus company?

Answer: 70 passengers

20x if 0 < x < 60

?

R = ?? (20 - 0.25(x - 60))x if 60 < x < 100

where x = # of passengers

9. The congruent sides of an isosceles triangle are each equal to 8 cm. Find the height of the triangle so that

its area is maximized.

Answer: h = 4 2

A=h

64 - h

2

10. An optical cable is to be laid under a straight 20-meter wide canal from point A to point C, and then

underground to point B which is 40 meters downstream from A. It costs $50 per meter to put the cable

underground and $75 per meter to lay the cable underwater. How far should point C be from B so that the cost

of laying down the cable is minimized?

Answer: 40 - 8 5 meters

Let x = BC.

Cost = 50x + 75

2

20 + (40-x)

2

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