A factory makes boxes of crayons. Each machine can make 100 boxes an hour. The cost of setting up a machine to make crayons is $50 per machine.

Use this information to answer the following:

1) The factory receives an order for 1000 boxes of crayons. Let t be the time spent making boxes of crayons, in hours, and m the number of machines used. Write an equation relating t and m. (Hint: the number of boxes is the product of the number of machines, the number of hours, and the number of boxes per hour per machine.)

2) Once the machines are set up, it only takes one person to monitor them. That person earns $30 per hour. Write the cost of making crayons with m machines for t hours as a sum of the cost of setting up the machines and the cost to pay the worker. Your answer will be in terms of m and t.

3) Combine your work in the previous two parts to write the cost of making 1000 boxes of crayons as a function only of the number m of machines.

4) Graph the function from the previous part using a graphing calculator or other graphing device and sketch the graph.

5) Use the graph to estimate the number of machines the company should use to minimize cost.

6) Suppose the factory sells the boxes of crayons for $5 each. Let P(m) be the amount of profit the factory makes if it makes 1000 boxes of chalk with m machines and sells all of them. Write a formula for P(m).

7) Graph the function P(m) using a graphing calculator or another graphing utility. Sketch what the graph looks like.

8) Use the graph to estimate the number of machines the company should use to maximize profit.

9) What do you notice about your answers to parts (4) and (7)? To parts (5) and (8)? Why are the answers to parts (5) and (8) related this way?