Profit with varying price: The background for this exercise can be found A manufacturer of widgets has fixed costs of $1200 per month, and the variable cost is $40 per widget (so it costs $40 to...

Profit with varying price: The background for this exercise can be found A manufacturer of widgets has fixed costs of $1200 per month, and the variable cost is $40 per widget (so it costs $40 to produce 1 widget). Let N be the number of widgets produced in a month.

a. Find a formula for the manufacturer’s total cost C as a function of N.

b. The highest price p, in dollars, of a widget at which N widgets can be sold is given by the formula p = 53 − 0.01N. Using this, find a formula for the total revenue R as a function of N.

c. Use your answers to parts a and b to find a formula for the profit P of this manufacturer as a function of N.

d. Use your formula from part c to determine the two break-even points for this manufacturer.Assume here that the manufacturer produces the widgets in blocks of 50, so a table setup showing N in multiples of 50 is appropriate.

e. Use your formula from part c to determine the production level at which profit is maximized if the manufacturer can produce at most 1500 widgets in a month. As in part d, assume that the manufacturer produces the widgets in blocks of 50.