To solve this problem, let's analyze each component step by step to understand how the profit function can be derived.
1. Operational Costs:
- The boathouse has a fixed operational cost of [tex]$\$[/tex] 2750[tex]$ each month.
- Additionally, for every boat that docks, the boathouse spends $[/tex]\[tex]$ 650$[/tex] per month.
2. Income:
- The boathouse charges a monthly fee of [tex]$\$[/tex] 900$ for each boat docked.
Let's define:
- [tex]\( n \)[/tex] as the number of boats.
3. Revenue Calculation:
- The total revenue from docking [tex]\( n \)[/tex] boats would be [tex]\( 900n \)[/tex] dollars.
4. Cost Calculation:
- The total monthly operational cost for [tex]\( n \)[/tex] boats is given by the fixed operational cost plus the cost per boat. Thus, the total cost would be:
[tex]\[
2750 + 650n
\][/tex]
5. Profit Calculation:
- Profit is the total revenue minus the total cost. Thus:
[tex]\[
\text{Profit} = \text{Total Revenue} - \text{Total Cost} = 900n - (2750 + 650n)
\][/tex]
6. Simplifying the Profit Equation:
- Let's distribute and combine like terms:
[tex]\[
\text{Profit} = 900n - 2750 - 650n
\][/tex]
- Combining the terms involving [tex]\( n \)[/tex]:
[tex]\[
\text{Profit} = (900n - 650n) - 2750 = 250n - 2750
\][/tex]
Therefore, the equation that represents the profit function of the boathouse is:
[tex]\[
C. \quad p = 250n - 2750
\][/tex]
So the correct answer is C.
