We have the cost function C(d) and the revenue function R(d) as:
[tex]C(d)=10500+3.9d[/tex][tex]\begin{gathered} R(d)=d(14.5-0.00003d) \\ R(d)=14.5d-0.00003d^2 \end{gathered}[/tex]
The profit can be defined as the difference between the revenue and the cost, so we can express it as:
[tex]\begin{gathered} P(d)=R(d)-C(d) \\ P(d)=(14.5d-0.00003d^2)-(10500+3.9d) \\ P(d)=-0.00003d^2+14.5d-3.9d-10500 \\ P(d)=-0.00003d+10.6d-10500 \end{gathered}[/tex]
We can now evaluate it for d = 30,000 as:
[tex]\begin{gathered} P=-0.00003\cdot(30,000)^2+10.6(30,000)-10,500 \\ P=-0.00003\cdot900,000,000+318,000-10,500 \\ P=-27,000+318,000-10,500 \\ P=280,500 \end{gathered}[/tex]
Answer: the profit for 30,000 units is $280,500 [Fourth option].
