To determine the profit function for the boathouse, we need to calculate the profit in terms of the number of boats, [tex]\( n \)[/tex], by considering both the income and the expenses. Let's break down the components one by one.
1. Monthly Operational Cost:
- The boathouse has a fixed operational cost of \[tex]$3000 per month. This cost is independent of the number of boats.
2. Monthly Cost for Each Boat:
- For each boat docked, the boathouse incurs an additional cost of \$[/tex]750 per month. For [tex]\( n \)[/tex] boats, the total cost is [tex]\( 750n \)[/tex].
3. Monthly Fee Charged for Each Boat:
- The boathouse charges a fee of \[tex]$950 per month per boat. For \( n \) boats, the total income is \( 950n \).
Now, let's compute the total cost and total revenue for \( n \) boats:
- Total Revenue (R):
- The revenue since the boathouse charges \$[/tex]950 per boat: [tex]\( R = 950n \)[/tex].
- Total Cost (C):
- The total cost includes the fixed operational cost plus the cost per boat:
[tex]\[ C = 3000 + 750n \][/tex]
The profit, [tex]\( p \)[/tex], is the total revenue minus the total cost:
[tex]\[ p = R - C \][/tex]
[tex]\[ p = 950n - (3000 + 750n) \][/tex]
Expanding the expression inside the parentheses and simplifying:
[tex]\[ p = 950n - 3000 - 750n \][/tex]
[tex]\[ p = (950n - 750n) - 3000 \][/tex]
[tex]\[ p = 200n - 3000 \][/tex]
Therefore, the equation that represents the profit function of the boathouse is:
[tex]\[ p = 200n - 3000 \][/tex]
Hence, the correct answer is:
- B. [tex]\( p = 200n - 3000 \)[/tex]
