On a tropical island there are 100 potential boat builders, numbered 1 through 100.Each can build up to20 boats a year, but anyone who goes into the boatbuilding business has to pay a fixed cost of $19.Marginal costs differ from person to person.Where y denotes the number of boats built per year, boat builder 1 has a total cost function c(y)= 19 + y.Boat builder 2 has a total cost function c(y)= 19 + 2y, and more generally, for each i, from 1 to 100, boat builder i has a cost function c(y)= 19 + iy.If the price of boats is 25,how many boats will be built per year?

Question

Quizplus · Accepted Answer

The cost function for each boat builder is c(y) = 19 + iy, where i is the boat builder's number and y is the number of boats built. To determine how many boats will be built, we need to find out for which boat builders the revenue (price per boat times the number of boats built) exceeds or equals their total cost. The price of boats is given as 25, so the revenue for building y boats is 25y. A boat builder will choose to build boats if 25y ≥ 19 + iy. Simplifying, we get 25y - iy ≥ 19, or y(25 - i) ≥ 19. Since each builder can build up to 20 boats, we need to find the highest boat builder number (i) for which building boats is profitable. Solving for i when y = 20 (the maximum number of boats a builder can make), we get 20(25 - i) ≥ 19, or 500 - 20i ≥ 19, simplifying to 481 ≥ 20i, and finally i ≤ 24.05. This means that boat builders numbered 1 through 24 will find it profitable to build boats, each producing up to 20 boats. Therefore, the total number of boats built per year is 24 builders * 20 boats/builder = 480 boats.

On a Tropical Island There Are 100 Potential Boat Builders