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A boathouse costs [tex]\[tex]$2750[/tex] a month to operate, and it spends [tex]\$[/tex]650[/tex] each month for every boat that it docks. The boathouse charges a monthly fee of [tex]\$900[/tex] to dock a boat. If [tex]n[/tex] is the number of boats, which equation represents the profit function of the boathouse?

A. [tex]p=650n + 2750[/tex]
B. [tex]p=2750n + 250[/tex]
C. [tex]p=250n - 2750[/tex]
D. [tex]p=900n - 2750[/tex]



Answer :

GinnyAnswer
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.

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