Question

A small independent motion picture company determines the profit for producing » DVDcopies of a recent release is P=-0.02n? + 3.40m - 16. Pis the profit in thousands of dollars and m isin thousands of units.B) What will the maximize profit be?

Answer 1

The Solution:

Given the profit function below:

`P=-0.02n^2+3.40n-16`

To maximize profit, we have to differentiate the given function P with respect to n, and then equate to zero.

`\begin{gathered} (dP)/(dx)=0 \\ \\ (dP)/(dx)=2(-0.02)n+3.40=0 \\ \\ -0.04n+3.40=0 \end{gathered}`

Solving for n, we get

`\begin{gathered} -0.04n=-3.40 \\ \text{ Dividing both sides by -0.04} \\ n=(-3.40)/(-0.04)=85\text{ thousand units of DVD} \end{gathered}`

Thus, the correct answer to part (a) is 85 thousand units of DVD.

Part (b)

We are required to find the maximum profit.

We shall substitute 85 for n in the given profit function.

`\begin{gathered} P(n)=-0.02n^2+3.40n-16 \\ \\ P(85)=-0.02(85)^2+3.40(85)-16 \end{gathered}`
`\begin{gathered} P(85)=128.5 \\ So, \\ The\text{ maximum profit is }128.5\text{ thousand of dollars.} \end{gathered}`

Therefore, the maximum profit is 128.5 thousand of dollars.