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.

Question

​(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