Question

A small independent motion picture company determines the profit Pfor producing n DVDcopies of a recent release is P =-0.02n2 + 3.40n - 16. Pis the profit in thousands of dollars and n isin thousands of units.Was it is question 5 ignore my notes

Answer 1

In calculus, at the maximum point, gradient is zero.

`\begin{gathered} We\text{ plot a graph of the curve P=}-0.02n^2\text{ +}3.40n\text{ - 16}\ldots\ldots.\ldots\ldots\text{.(eqn 1)} \\ \text{Gradient of the curve at any point is }(dP)/(dn)=\text{-}0.04n\text{ +3.40} \end{gathered}`

Recall that at the maximum point, this dP/dn is zero and in that case, we have:

`\begin{gathered} 0\text{ = -0.04n + 3.40 } \\ \text{adding 0.04n to both sides give} \\ 0.04n\text{ = 3.40} \\ \text{dividing both sides by 0.04} \\ (0.04n)/(0.04)=(3.40)/(0.04) \\ n\text{ = 85 thousand units.} \end{gathered}`
`\begin{gathered} \text{Next step is to substitute n into the equation for P (eqn 1)} \\ P=-0.02(85^2)\text{ + 3.4(85) - 16 = 128}.5\text{ thousand dollars.} \end{gathered}`

Therefore the company will produce 85 thousand units to maximize profits

and in producing these, will make its maximum profit which is 128.5 thousand dollars