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]
