a film company is deciding on the price of the video release of its film cousin of frankenstein. its…

a film company is deciding on the price of the video release of its film cousin of frankenstein. its marketing people estimate that at a price of p dollars, it can sell a total of q = 200000 - 10000p copies. what price will bring in the greatest revenue? the optimal price is p = dollars.

Answer

Explanation:

Step1: Define the revenue function

Revenue $R = p\times q$. Given $q = 200000 - 10000p$, then $R(p)=p(200000 - 10000p)=200000p-10000p^{2}$.

Step2: Find the derivative of the revenue function

The derivative $R^\prime(p)=\frac{d}{dp}(200000p - 10000p^{2})=200000-20000p$.

Step3: Set the derivative equal to zero to find critical points

Set $R^\prime(p)=0$, so $200000 - 20000p=0$. Solving for $p$ gives $20000p = 200000$, then $p = 10$.

Step4: Check the second - derivative to confirm it's a maximum

The second - derivative $R^{\prime\prime}(p)=\frac{d}{dp}(200000 - 20000p)=-20000<0$. Since $R^{\prime\prime}(p)<0$, when $p = 10$, the revenue function $R(p)$ has a maximum.