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.

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?
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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 the problem, we will use the profit function given:

\[ P(x) = 20x - 6000 \]

Part 1: Find the number of hats for a profit of $8000

We set the profit function equal to $8000 and solve for \( x \):

\[ 8000 = 20x - 6000 \]

First, add 6000 to both sides:

\[ 8000 + 6000 = 20x \]

\[ 14000 = 20x \]

Next, divide by 20:

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

Answer for Part 1: Producing and selling 700 hats will give a profit of $8000.

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Part 2: Find the number of hats to avoid a loss

To avoid a loss, the profit must be at least $0. We set the profit function to be greater than or equal to $0:

\[ 0 \leq 20x - 6000 \]

Add 6000 to both sides:

\[ 6000 \leq 20x \]

Next, divide by 20:

\[ x \geq \frac{6000}{20} = 300 \]

Answer for Part 2: To avoid a loss, 300 hats must be produced and sold.