Alright, let's work through this problem step-by-step.
### Part (a): Write the Profit Function
1. Given Functions:
- Cost function [tex]\( C(x) = 115x + 60,000 \)[/tex]
- Revenue function [tex]\( R(x) = 235x \)[/tex]
2. Profit Function:
The profit function [tex]\( P(x) \)[/tex] is the difference between the revenue function [tex]\( R(x) \)[/tex] and the cost function [tex]\( C(x) \)[/tex].
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
3. Substitute the given functions:
[tex]\[
P(x) = 235x - (115x + 60,000)
\][/tex]
4. Simplify the expression:
[tex]\[
P(x) = 235x - 115x - 60,000
\][/tex]
[tex]\[
P(x) = 120x - 60,000
\][/tex]
So, the profit function is:
[tex]\[
P(x) = 120x - 60,000
\][/tex]
### Part (b): Determine the Break-Even Point
We need to determine more than how many units must be produced and sold for the business to start making money, i.e., where the profit [tex]\( P(x) \)[/tex] is greater than zero.
1. Set the profit function greater than zero:
[tex]\[
120x - 60,000 > 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
120x > 60,000
\][/tex]
[tex]\[
x > \frac{60,000}{120}
\][/tex]
[tex]\[
x > 500
\][/tex]
Therefore, more than 500 units must be produced and sold for the business to start making a profit.
### Summary
- Profit function [tex]\( P(x) \)[/tex]:
[tex]\[
P(x) = 120x - 60,000
\][/tex]
- Break-even point:
More than 500 units must be produced and sold for the business to make money.
