1. A toll road commission is planning to locate garages for tow trucks along a 100-kilometre circular highway. Each garage has a fixed cost of 5,000 per day.Towing jobs are equally likely along any point of the highway, and cost per kilometre towedis 50. If there were 5,000 towing jobs per day, what number of garages wouldminimize the sum of the fixed costs and towing costs?

ANSWER: Suppose we have N garages. Following the example in the text, the distance between two adjacent garages will be 100/N. The average towing distance will thus be 100/4N. Average round-trip distance is 100/2N.

1. There are 5000 towing incidents with a cost of50/km each. Total direct towing cost is thus (5000)(50)(100/2N)=12,500,000/N. Total cost of the garages is 5000N.Overall total cost is TC = 5000N + 12,500,000/N.
2. To find the optimal N, we could graph this function. Alternatively, taking the first derivative and setting it to zero will give the minimum. dTC/dN = 5000 - 12,500,000/N2= 0. This solves for N = 50.

I have struggled to understand how the formula was used, and I am not quite sure about it, please explain it step by step, thanks a lot!