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You decide to begin selling bags of peanuts at the local swap mart. Your cost for each bag of peanuts is [tex]$1.59, and you have to pay a fixed weekly fee of $[/tex]120 for the booth. You plan to sell each bag of peanuts for $3.05.

a. Write a function, [tex]\(C(n)\)[/tex], to represent your total costs for the week if you sell [tex]\(n\)[/tex] bags of peanuts.
[tex]\[C(n) = \][/tex]

b. Write a function, [tex]\(R(n)\)[/tex], to represent the revenue from the sale of [tex]\(n\)[/tex] bags of peanuts during the week.
[tex]\[R(n) = \][/tex]

c. Write a function, [tex]\(P(n)\)[/tex], that represents the profits for selling [tex]\(n\)[/tex] bags of peanuts in a given week.
[tex]\[P(n) = \][/tex]

d. How many items must you sell to break even?
[tex]\[\_\_\_\_\_ \text{ bags of peanuts}\][/tex]

Sagot :

GinnyAnswer
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]
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