A manufacturer of tennis rackets finds that the total cost C(x) (in dollars) of manufacturing x rackets/day is given by Each racket can be sold at a price of p dollars, where p is related to x by the demand equation If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer.
A) 6,000 rackets/day
B) 11,000 rackets/day
C) 7,000 rackets/day
D) 8,000 rackets/day

Question

A manufacturer of tennis rackets finds that the total cost C(x) (in dollars) of manufacturing x rackets/day is given by  Each racket can be sold at a price of p dollars, where p is related to x by the demand equation  If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer.
A) 6,000 rackets/day 
B) 11,000 rackets/day 
C) 7,000 rackets/day 
D) 8,000 rackets/day

Quizplus · Accepted Answer

The profit function is given by:

P(x) = R(x) - C(x)

where R(x) is the revenue function:

R(x) = xp(x) = x(35 - 0.002x)

Substituting the cost function, we get:

P(x) = x(35 - 0.002x) - 8000 - 12x + 0.001x^2

P(x) = -0.001x^2 + 23x - 8000

To maximize profit, we take the derivative of P(x) and set it to zero:

P'(x) = -0.002x + 23 = 0

Solving for x, we get:

x = 11,500

Since the production level has to be a whole number, we choose the nearest whole number, which is:

x = 11,000

Therefore, the daily level of production that will yield a maximum profit for the manufacturer is 11,000 rackets/day.

A Manufacturer of Tennis Rackets Finds That the Total Cost