1. The basketball team is selling wreaths during the winter to raise money for a trip. The overhead to make the wreaths is [tex]\$150[/tex] plus [tex]\$10[/tex] per wreath. They are selling the wreaths for [tex]\$20[/tex].

The total cost function for producing the wreaths is [tex]C = 150 + 10n[/tex], where [tex]n[/tex] is the number of wreaths. The revenue is [tex]R = 20n[/tex], where [tex]n[/tex] is the number of wreaths.

If the profit [tex](P)[/tex] is given by [tex]P = R - C[/tex], what is the profit function for this situation?

A. [tex]P = 30n - 150[/tex]

B. [tex]P = 150 - 10n[/tex]

C. [tex]P = 10n - 150[/tex]

D. [tex]P = 150 + 30n[/tex]



Answer :

To determine the profit function [tex]\( P \)[/tex] for the basketball team's wreath sales, we follow these steps:

1. Identify the Cost Function [tex]\( C \)[/tex]:

The overhead to make the wreaths is [tex]$150$[/tex] plus [tex]$10$[/tex] per wreath. This can be described by the cost function:
[tex]\[ C = 150 + 10n \][/tex]
where [tex]\( n \)[/tex] is the number of wreaths produced.

2. Identify the Revenue Function [tex]\( R \)[/tex]:

The revenue from selling the wreaths is $20 per wreath, described by the revenue function:
[tex]\[ R = 20n \][/tex]
where [tex]\( n \)[/tex] is again the number of wreaths sold.

3. Write the Profit Function [tex]\( P \)[/tex]:

Profit is defined as the difference between revenue and cost:
[tex]\[ P = R - C \][/tex]

Substituting the given expressions for [tex]\( R \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ P = 20n - (150 + 10n) \][/tex]

4. Simplify the Expression:

Distribute the negative sign through the parentheses:
[tex]\[ P = 20n - 150 - 10n \][/tex]

Combine like terms:
[tex]\[ P = (20n - 10n) - 150 \][/tex]
[tex]\[ P = 10n - 150 \][/tex]

Therefore, the profit function [tex]\( P \)[/tex] for the basketball team's wreath sales is:
[tex]\[ P = 10n - 150 \][/tex]

So, the correct answer from the given options is:
[tex]\[ \boxed{P = 10n - 150} \][/tex]