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

To find the daily level of production that will yield a maximum profit, we need to determine the value of x that maximizes the profit function:

P(x) = R(x) - C(x)
P(x) = xp(x) - 11eb12b0_a9e5_1a78_b394_cd918b7a3dd2_TB7866_11

Substituting the given demand equation for p(x), we get:

P(x) = x(2000 - 50x) - 11eb12b0_a9e5_1a78_b394_cd918b7a3dd2_TB7866_11

Expanding and simplifying, we get:

P(x) = -50x^2 + 2000x - 11eb12b0_a9e5_1a78_b394_cd918b7a3dd2_TB7866_11

To find the value of x that maximizes this quadratic function, we can take the derivative with respect to x and set it equal to zero:

P'(x) = -100x + 2000 = 0
x = 20

Since the quadratic function is concave down (i.e. the coefficient of x^2 is negative), this critical point corresponds to a maximum. Therefore, the daily level of production that will yield a maximum profit is:

x = 20

Substituting this back into the demand equation, we can find the corresponding price and profit:

p = 2000 - 50x = 1000 dollars/racket
P = xp - C = 20(1000) - 11eb12b0_a9e5_1a78_b394_cd918b7a3dd2_TB7866_11 = 189eb12b0_a9e5_1a78_b394_cd918b7a3dd2_TB7866_11

Therefore, the manufacturer should produce 8,000 rackets/day to maximize their profit.

A Manufacturer of Tennis Rackets Finds That the Total Cost