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:

$$20x-2000\ge 0$$

Now, solve for $x$:

1. Add 2000 to both sides: $$20x\ge 2000$$

2. Divide by 20: $$x\ge \frac{2000}{20}$$ $$x\ge 100$$

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

Final Answers:

• (a) 500 hats
• (b) 100 hats