Question

The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total cost, we need to know the manufacturer's fixed costs (covering things such as plant maintenance and insurance), as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs.

The total revenue R for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total revenue, we need to know the selling price per unit of the item. To find the total revenue, we multiply this selling price by the number of items produced.

The profit P for a manufacturer is the total revenue minus the total cost. If this number is positive, then the manufacturer turns a profit, whereas if this number is negative, then the manufacturer has a loss. If the profit is zero, then the manufacturer is at a break-even point.

Suppose that a manufacturer of widgets has fixed costs of $9000 per month and that the variable cost is $15 per widget (so it costs $15 to produce 1 widget). Suppose the manufacturer of widgets sells the widgets for $21 each.

(a) Use a formula to express this manufacturer's total revenue R in a month as a function of the number of widgets produced in a month. (Let N be the number of widgets produced each month.)
R = _____________

(b) Use a formula to express the profit P of this manufacturer as a function of the number of widgets produced in a month. (Let N be the number of widgets produced each month.)
P = ______________

(c) Express using functional notation the profit P of this manufacturer if there are 200 widgets produced in a month, and then calculate that value.
P( ______ ) = $ ________
(d) At the production level of 200 widgets per month, does the manufacturer turn a profit or have a loss?

Is it a profit, loss, or break even?

What about the production level of 1000 widgets per month?

Is it a profit, loss, or break even?