Question: A printer has a contract to print 100,000 invitations for a political candidate. He can run the invitations by using any number of metal printing plates from 1 to 20 on his press. For example, if he prepares 10 plates, each impression of his press makes 10 invitations. Preparing each plate costs $8 and he can make 1000 impressions per plate per hour. If it costs $128 per hour to run the press, how many metal printing plates should the printer use to minimize the cost of printing the 100,000 invitations?

A. To understand the problem:

i. Determine how many hours it would take to print the 100,000 invitations with 10 plates.

ii. How much would it cost to print the invitations with 10 plates?

iii. Determine how much it would cost to prepare 10 printing plates.

iv. What is the total cost to make the 10 plates and print 100,000 invitations?

B. To find the number of plates that will minimize the total cost of printing the invitations:

i. Find the cost of preparing x plates.

ii. Find the number of invitations that can be made with one impression made by the x plates.

iii. Find the number of invitations that can be made per hour with x plates and the number of hours it would take to print 100,000 invitations.

iv. Write a function of x that gives the total cost of producing the invitations.

v. Use technology to graph the cost function and to estimate the number of plates that will give the minimum cost of producing 100,000 plates.

vi. Use numerical methods to find the number of plates that will give the minimum cost and the minimum possible cost of producing 100,000 invitations.