Suppose that a sporting goods company manufactures basketballs and soccer balls

Question

Suppose that a sporting goods company manufactures basketballs and soccer balls in quantities q_1 and q_2, respectively, and that it sells them at prices p_1 and p_2, respectively. Suppose that the company's production costs are given by C = 2q_1^2 + 2q_2^2 + 10. Find the maximum profit that the company can make assuming that prices are fixed. Find the rate of change of the maximum profit you found in part as the price p_1 increases. Is it wise for the company to increase the price of basketballs?

Explanation / Answer

C=2q1^2 +2q2^2 +10

profit=revenue -cost

revenue =p1q1+p2q2

profit=p1q1+p2q2-2q1^2 -2q2^2 -10

d(profit)/dq1=p1-4q1

d(profit)/dq2=p2-4q2

d(profit)/dq1=d(profit)/dq2=0 for maximum profit

q1=p1/4

q2=p2/4

so maximum profit is

p1q1+p2q2-2q1^2 -2q2^2 -10

=p1*p1/4+p2*p2/4 -2(p1/4)^2 -2(p2/4)^2-10

=(p1^2)/4 +(p2^2)/4 -(p1^2)/8 -(p2^2)/8 -10

p(max)=(p1^2)/8 +(p2^2)/8 -10

b)dp(max)/dp1 =(p1)/4

as p1 increases rate of change of maximum profit also increases.

Yes its wise to increase price of basketballs.

Thank you

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