Let's go through each part step-by-step to provide detailed solutions for the questions.
### Part (a) - Cost Function [tex]\( C(n) \)[/tex]:
To find the total cost [tex]\( C(n) \)[/tex] for selling [tex]\( n \)[/tex] bags of peanuts, we need to consider both the variable costs and the fixed costs:
- Variable cost per bag: [tex]$1.59$[/tex]
- Fixed weekly fee: [tex]$120
The total cost function \( C(n) \) can be written as:
\[
C(n) = 1.59n + 120
\]
### Part (b) - Revenue Function \( R(n) \):
To find the total revenue \( R(n) \) from selling \( n \) bags of peanuts, use the sale price per bag, which is $[/tex]3.05:
[tex]\[
R(n) = 3.05n
\][/tex]
### Part (c) - Profit Function [tex]\( P(n) \)[/tex]:
The profit [tex]\( P(n) \)[/tex] can be calculated by subtracting the total costs [tex]\( C(n) \)[/tex] from the total revenue [tex]\( R(n) \)[/tex]:
[tex]\[
P(n) = R(n) - C(n) = 3.05n - (1.59n + 120) = 3.05n - 1.59n - 120 = 1.46n - 120
\][/tex]
### Part (d) - Break-Even Point:
To find the number of bags of peanuts needed to break even, we set the total revenue equal to the total costs:
[tex]\[
R(n) = C(n)
\][/tex]
[tex]\[
3.05n = 1.59n + 120
\][/tex]
Subtract [tex]\( 1.59n \)[/tex] from both sides:
[tex]\[
3.05n - 1.59n = 120
\][/tex]
[tex]\[
1.46n = 120
\][/tex]
Divide both sides by 1.46:
[tex]\[
n = \frac{120}{1.46} \approx 82.19
\][/tex]
Therefore, you need to sell approximately [tex]\( 82.19 \)[/tex] bags of peanuts to break even.
### Summarized Answer:
- (a) Cost function:
[tex]\[
C(n) = 1.59n + 120
\][/tex]
- (b) Revenue function:
[tex]\[
R(n) = 3.05n
\][/tex]
- (c) Profit function:
[tex]\[
P(n) = 1.46n - 120
\][/tex]
- (d) Number of items to sell to break even:
[tex]\[
\approx 82.19 \text{ bags of peanuts}
\][/tex]
