Cheaper by the Dozen (30 points total) For the following questions, assume that: • Total initial costs are $3800. • You can expect to sell 100 items per week if the price is $5 per item. • Sales per week increase by 10 items for every $1 you lower the price. 31. The cost to manufacture one item often decreases as you make more items. For instance, the company providing your printer supplies may give price breaks for larger orders. Suppose that your per-item cost is $3.00, minus $0.01 for every item you manufacture that week. Some examples: Items manufactured 1 10 50 100 Cost per item $2.99 $2.90 $2.50 $2.00 Write an expression for the cost per item when manufacturing x items. Be sure your expression gives the same results as the table. (4 points) 32. Using the cost-per-item expression from question 31, write the cost function C(x) for manufacturing x items. Remember to include the initial costs. Clear any parentheses by using the distributive property. (8 points: 4 points for the initial expression and 4 points for the simplified form)

You want an expression for cost per item and one for the manufacturing cost of x items, given the per-item cost is $3.00, minus $0.01 for every item, and the fixed manufacturing costs are $3800.

31. Cost per item

We can express the description "$3.00, minus $0.01 for every item" as the expression ...

3.00 -0.01x

32. Cost function

The cost function will be the total of initial costs and the product of the number of items and the cost per item:

C(x) = 3800 +x(3.00 -0.01x)

C(x) = 3800 +3.00x -0.01x² . . . . . . . simplified

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Additional comment

The number sold at price p is q(p) = 100 +10(5 -p). This is maximized at 75 when the price is 7.50. However, maximum profit is had when 67 are sold at a price of $8.30. This results in weekly profit of $399.99, not enough to cover the initial cost.

Hence the cost function isn't quite right, as it mixes a one-time cost with the per-week cost. It would seem to take just over 9.5 weeks to make enough profit to cover the initial costs.