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?

Answer 1

To solve the profit function \( P(x) = 20x - 6000 \) for the scenarios provided:

(a) To find how many hats will give a profit of $8000:

Set \( P(x) \) equal to 8000:

\[ 20x - 6000 = 8000 \]

Now, solve for \( x \):

1. Add 6000 to both sides: \[ 20x = 8000 + 6000 \] \[ 20x = 14000 \]

2. Divide by 20: \[ x = \frac{14000}{20} = 700 \]

So, producing and selling 700 hats will give a profit of $8000.

(b) To 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:

\[ 20x - 6000 \geq 0 \]

Now, solve for \( x \):

1. Add 6000 to both sides: \[ 20x \geq 6000 \]

2. Divide by 20: \[ x \geq \frac{6000}{20} = 300 \]

Thus, to avoid a loss, at least 300 hats must be produced and sold.