Sure! Let's break down the problem step by step.
### Part (a)
To find the number of hats needed to achieve a profit of [tex]$8000, we need to set \( P(x) \) equal to $[/tex]8000 and solve for [tex]\( x \)[/tex]. The given profit function is:
[tex]\[ P(x) = 20x - 6000 \][/tex]
We set this equal to [tex]$8000:
\[ 20x - 6000 = 8000 \]
Now, solve for \( x \):
1. Add 6000 to both sides of the equation:
\[ 20x = 8000 + 6000 \]
\[ 20x = 14000 \]
2. Divide both sides by 20:
\[ x = \frac{14000}{20} \]
\[ x = 700 \]
So, producing and selling 700 hats will give a profit of $[/tex]8000.
### Part (b)
To avoid a loss, the profit must be at least zero. This means we need to find the number of hats [tex]\( x \)[/tex] for which [tex]\( P(x) = 0 \)[/tex]. Set the profit function equal to zero:
[tex]\[ 20x - 6000 = 0 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
1. Add 6000 to both sides of the equation:
[tex]\[ 20x = 6000 \][/tex]
2. Divide both sides by 20:
[tex]\[ x = \frac{6000}{20} \][/tex]
[tex]\[ x = 300 \][/tex]
So, producing and selling at least 300 hats will avoid a loss.
### Summary
(a) Producing and selling 700 hats will give a profit of $8000.
(b) Producing and selling at least 300 hats will avoid a loss.
