Question
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.
(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.
(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.
Answers
GPT-4o mini
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.
---
### 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.
\[ P(x) = 20x - 6000 \]
### 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.
---
### 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.