Question

The profit from the production and sale of specialty golf hats is given by the function ​P(x)equals20xminus6000 where x is the number of hats produced and sold.
​(a) Producing and selling how many hats will give a profit of ​$8000​?
​(b) How many hats must be produced and sold to avoid a​ loss?

Answer 1

To solve the profit function \( P(x) = 20x - 6000 \) for the scenarios provided:

### (a) To find how many hats will give a profit of $8000:

Set \( P(x) \) equal to 8000:

\[
20x - 6000 = 8000
\]

Now, solve for \( x \):

1. Add 6000 to both sides:
\[
20x = 8000 + 6000
\]
\[
20x = 14000
\]

2. Divide by 20:
\[
x = \frac{14000}{20} = 700
\]

So, producing and selling **700 hats** will give a profit of **$8000**.

### (b) To find how many hats must be produced and sold to avoid a loss:

To avoid a loss, the profit \( P(x) \) must be greater than or equal to 0:

\[
20x - 6000 \geq 0
\]

Now, solve for \( x \):

1. Add 6000 to both sides:
\[
20x \geq 6000
\]

2. Divide by 20:
\[
x \geq \frac{6000}{20} = 300
\]

Thus, to avoid a loss, at least **300 hats** must be produced and sold.