The manufacturer has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. (a) Verify that the formula p = 50 − 0.04N, where p is the price in dollars, gives the same values as those in the table. Use the formula to fill in the missing values for p. N = Number of widgets sold p = Price 100 46 200 42 300   400 34 500   (b) Use the formula from part (a) and the fact that R is the product of p and N to find a formula expressing the total revenue R as a function of N for this widget manufacturer. R =        (c) Express using functional notation the total revenue of this manufacturer if there are 375 widgets produced in a month, and then calculate that value. (Round your answer to the nearest cent.) R(  ) = $

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.

In general, the highest price p per unit of an item at which a manufacturer can sell N items is not constant but is rather a function of N. The total revenue R is still the product of p and N, but the formula for R is more complicated when p depends on N. The manufacturer has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold.