Question

The profit that a company makes selling an item (in thousands ofdollars) depends on the price of the item (in dollars). If p is the price ofthe item, then three equivalent forms for the profit are:Standard form: :-2p? +24p-54Factored form: -2(p-3)(p-9)Vertex form:-2(p-6?+1S.

Answer 1

Explanation

We are given the function that defines the profit in relation to the price of the item

The first part of the question:

The form that is most useful for finding the price that gives a profit of 0 dollars will be

the standard form because we can easily substitute p =0 and get the profit

So we have the answer to be

$-2p^2+24p-54$

For the second part of the question

The profit, when the price is 0 will be when p =0

so that

$\begin{gathered} profit=-2(0)^2+24(0)-54=0-54=-54 \\ \\ Since\text{ the unit of the profit is thousand dollar, then the profit will be} \\ -\text{ \ 54,000} \end{gathered}$

The profit is -$54,000

The third part of the question

The price that gives maximum profit can simply be obtained from the vertex equation

$-2(p-6)^2+18$

This will be when

$\begin{gathered} p-6=0 \\ p=6 \end{gathered}$

So the price that gives maximum profit is $6