Question

The profit from the production and sale of specialty golf hats is given by the function ​P(x)equals20xminus7000 where x is the number of hats produced and sold.
​(a) Producing and selling how many hats will give a profit of ​$7000​?
​(b) How many hats must be produced and sold to avoid a​ loss?
Question content area bottom
Part 1
​(a) Producing and selling
  
enter your response here hats will give a profit of ​$7000.
Part 2
​(b) To avoid a​ loss,
  
enter your response here hats must be produced and sold.

Answer 1

To solve both parts of your problem, we'll use the profit function \( P(x) = 20x - 7000 \).

### Part 1: Find how many hats give a profit of $7000.

We set the profit \( P(x) \) equal to 7000:

\[
20x - 7000 = 7000
\]

Now, we solve for \( x \):

1. Add 7000 to both sides:

\[
20x = 7000 + 7000
\]

\[
20x = 14000
\]

2. Divide both sides by 20:

\[
x = \frac{14000}{20} = 700
\]

Thus, **producing and selling 700 hats will give a profit of $7000**.

### Part 2: 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:

\[
20x - 7000 \geq 0
\]

Now, solve for \( x \):

1. Add 7000 to both sides:

\[
20x \geq 7000
\]

2. Divide both sides by 20:

\[
x \geq \frac{7000}{20} = 350
\]

Thus, **to avoid a loss, at least 350 hats must be produced and sold**.

### Final Answers:
- Part (a): **700 hats**
- Part (b): **350 hats**