the equation below is an approximation to the total profit (in thousands of dollars) from the sale of x…

Explanation:

Step1: Find the derivative

Differentiate $P(x)=-x^{3}+\frac{27}{2}x^{2}-60x + 100$ using power - rule. $P'(x)=-3x^{2}+27x - 60$.

Step2: Set derivative equal to 0

Solve $-3x^{2}+27x - 60 = 0$. Divide through by - 3: $x^{2}-9x + 20=0$.

Step3: Factor the quadratic

Factor $x^{2}-9x + 20$ to get $(x - 4)(x - 5)=0$. So $x = 4$ or $x = 5$. But since $x\geq5$, we discard $x = 4$.

Step4: Check the second - derivative

Differentiate $P'(x)$ to get $P''(x)=-6x + 27$. When $x = 5$, $P''(5)=-6\times5+27=-3<0$, so $x = 5$ is a maximum.

Answer:

Since $x$ is in hundreds of thousands of tires, the number of tires is $5\times100000 = 500000$, so D. 500,000.