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:

$\begin{gathered} 6000=20x-8000 \\ solve_{\text{ }}for_{\text{ }}x\colon \\ 20x=6000+8000 \\ 20x=14000 \\ x=\frac{14000}{20} \\ x=700 \end{gathered}$

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

-----------------------

(b)

P(x) ≥ 0

So:

$\begin{gathered} 20x-8000\ge0 \\ solve_{\text{ }}for_{\text{ }}x\colon \\ 20x-8000\ge0 \\ 20x\ge8000 \\ x\ge\frac{8000}{20} \\ x\ge400 \end{gathered}$

To avoid a loss 400 hats must be produced and sold