Question

The following information is relevant for Questions 8-9. The rope tying a barge to the dock snapped; the barge began floating down the river, and struck a small boat. Suppose that the barge operator can reduce the probability of the rope snapping by using a thicker rope. Let x be the thickness of the rope and p(x) = 1/x be the probability of snapping. Moreover, suppose that the thickness of the rope can vary between 5 and 15 inches (with 1 inch increments). Assume that the rope costs 1000 per inch of thickness (for the length of rope needed) and that the accident produces 100000 worth of damage.Bottom of Form

Question 8

HomeworkUnanswered

What is the total cost of choosing a rope with thickness x (note: total cost accounts for both the expected cost of the accident as well as the precaution cost)?

A 100000/x

B 100000/x+1000

C 100000/x+1000x

D 100000+1000x

Question 9

HomeworkUnanswered

The optimal thickness of the rope that minimizes total cost is

A 5

B 8

C 10

D 12

E 14

F 15

Answer 1

Question 8

Cost of rope of thickness x=C₁=1000*x

Expected cost of snapping=C₂=p(x)*Loss amount=(1/x)*100000=100000/x

Total Cost=TC =C₁+C₂=1000x+(100000/x)

Correct option is

C 100000/x+1000x

Question 9

We have determined that

TC=1000x+(100000/x)

Differentiate TC with respect to x, we get

dTC/dx=1000-(100000/x²)

Set dTC/dx=0 for cost minimization

1000-(100000/x²)=0

1000x²=100000

x²=100

x=10 inches (optimal thickness)

Correct option is

C. x=10