Answer :
Sure, let's solve the problem step-by-step.
The problem states that there is a maintenance cost of \[tex]$250,000 per year for the apartment complex. Additionally, there are 500 apartments that will be rented out for a monthly rental fee.
We need to establish a profit function \(P(x)\) based on the monthly rent \(x\).
Here's how we will proceed:
1. Determine the annual rental income: Each apartment generates \(x\) dollars per month. With 500 apartments, the monthly rental income is \(500 \times x\). Since there are 12 months in a year, the annual rental income becomes \(500 \times x \times 12\).
2. Subtract the maintenance cost: The annual cost to maintain the apartment complex is \$[/tex]250,000. The profit function will thus be the annual rental income minus this maintenance cost.
Now let's put the steps together mathematically:
- Annual rental income = [tex]\(500 \times x \times 12\)[/tex]
- Maintenance cost = \$250,000
Profit function [tex]\(P(x)\)[/tex] = Annual rental income - Maintenance cost
[tex]\[ P(x) = (500 \times x \times 12) - 250000 \][/tex]
[tex]\[ P(x) = 6000x - 250000 \][/tex]
We compare this with the given options:
1. [tex]\( P(x) = 500x + 250000 \)[/tex]
2. [tex]\( P(x) = 6000x - 250000 \)[/tex]
3. [tex]\( P(x) = 250000 + 6000x \)[/tex]
4. [tex]\( P(x) = 50000x \)[/tex]
From our derived function, it's clear that the correct profit function matches:
[tex]\[ P(x) = 6000x - 250000 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{P(x) = 6000x - 250000} \][/tex]
The problem states that there is a maintenance cost of \[tex]$250,000 per year for the apartment complex. Additionally, there are 500 apartments that will be rented out for a monthly rental fee.
We need to establish a profit function \(P(x)\) based on the monthly rent \(x\).
Here's how we will proceed:
1. Determine the annual rental income: Each apartment generates \(x\) dollars per month. With 500 apartments, the monthly rental income is \(500 \times x\). Since there are 12 months in a year, the annual rental income becomes \(500 \times x \times 12\).
2. Subtract the maintenance cost: The annual cost to maintain the apartment complex is \$[/tex]250,000. The profit function will thus be the annual rental income minus this maintenance cost.
Now let's put the steps together mathematically:
- Annual rental income = [tex]\(500 \times x \times 12\)[/tex]
- Maintenance cost = \$250,000
Profit function [tex]\(P(x)\)[/tex] = Annual rental income - Maintenance cost
[tex]\[ P(x) = (500 \times x \times 12) - 250000 \][/tex]
[tex]\[ P(x) = 6000x - 250000 \][/tex]
We compare this with the given options:
1. [tex]\( P(x) = 500x + 250000 \)[/tex]
2. [tex]\( P(x) = 6000x - 250000 \)[/tex]
3. [tex]\( P(x) = 250000 + 6000x \)[/tex]
4. [tex]\( P(x) = 50000x \)[/tex]
From our derived function, it's clear that the correct profit function matches:
[tex]\[ P(x) = 6000x - 250000 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{P(x) = 6000x - 250000} \][/tex]
