he profit from the production and sale of specialty golf hats is given by the function ​P(x)equals20xminus2000 where x is the number of hats produced and sold.

Question

​(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?
Question content area bottom
Part 1
​(a) Producing and selling
  
enter your response here hats will give a profit of ​$8000.
Part 2
​(b) To avoid a​ loss,
  
enter your response here hats must be produced and sold.

Answer 1

To solve for the answers, we will use the profit function \( P(x) = 20x - 2000 \).

Part (a): Find the number of hats to produce and sell for a profit of $8000.

We set the profit function equal to $8000:

\[ 20x - 2000 = 8000 \]

Now, solve for \( x \):

1. Add 2000 to both sides: \[ 20x = 8000 + 2000 \] \[ 20x = 10000 \]

2. Divide by 20: \[ x = \frac{10000}{20} \] \[ x = 500 \]

Therefore, producing and selling 500 hats will give a profit of $8000.

Part (b): Find the number of hats to produce and sell to avoid a loss.

To avoid a loss, we need the profit to be greater than or equal to 0:

\[ 20x - 2000 \geq 0 \]

Now, solve for \( x \):

1. Add 2000 to both sides: \[ 20x \geq 2000 \]

2. Divide by 20: \[ x \geq \frac{2000}{20} \] \[ x \geq 100 \]

Therefore, to avoid a loss, 100 hats must be produced and sold.

Final Answers:

• (a) 500 hats
• (b) 100 hats