Question

The profit from the production and sale of specialty golf hats is given by the function P(x)=20x−8000 where x is the number of hats produced and sold.(a) Producing and selling how many hats will give a profit of $6000?(b) How many hats must be produced and sold to avoid a loss?

Answer 1

The function is given by:

`P(x)=20x-8000`

(a) P(x) = 6000

so:

`\begin{gathered} 6000=20x-8000 \\ solve_{\text{ }}for_{\text{ }}x\colon \\ 20x=6000+8000 \\ 20x=14000 \\ x=(14000)/(20) \\ x=700 \end{gathered}`

Producing and selling 700 hats will give a profit of $6000

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(b)

P(x) ≥ 0

So:

`\begin{gathered} 20x-8000\ge0 \\ solve_{\text{ }}for_{\text{ }}x\colon \\ 20x-8000\ge0 \\ 20x\ge8000 \\ x\ge(8000)/(20) \\ x\ge400 \end{gathered}`

To avoid a loss 400 hats must be produced and sold