Question

The revenue generated from a 80 unit apartment building can be modeled with the polynomialR= - 28x^2 + 360x+ 29000 where x is the number of apartments rented at one time. The monthlycost to maintain the apartments is C = 300x+ 1100, again x is equal to the number of apartmentsrented.Find the profit polynomial that describes the total profit generated by the apartment complex based on thenumber of apartments rented.Give the profit earned if 40 units are rented.

Answer 1

The profit polynomial that describes the total profit generated by the apartment complex based on the number of apartments rented is
`P(x)=- 28x^2 + 60x+ 27900`.

The profit earned if 40 units are rented is 0.

In Mathematics and Financial accounting, profit is a measure of the amount of money generated when the selling price is deducted from the cost price of a good or service, which is usually provided by producers.

Mathematically, the amount of profit made from selling x bracelets can be calculated by using the following function:

Profit made = Revenue earned – Cost of production

P(x) = R(x) - C(x)

By substituting the given parameters, we have the following;

`P(x)=- 28x^2 + 360x+ 29000-(300x+ 1100)\\\\P(x)=- 28x^2 + 360x+ 29000-300x- 1100\\\\P(x)=- 28x^2 + 60x+ 27900`

When x is 40 units, the profit is as follows;

`P(x)=- 28x^2 + 60x+ 27900\\\\P(40)=- 28(40)^2 + 60(40)+ 27900`

P(40) = -14500

Therefore, the profit is 0 because it cannot assume a negative value and as such, represents a loss.

Answer 2

Given:

`\begin{gathered} R=-28x^2+360x+29000 \\ C=300x+1100 \\ \\ x\text{ is the number of apartments} \end{gathered}`

Profit is the surplus left from revenue after paying all costs.

Profit is found by deducting total costs from revenue.

Therefore, profit = total revenue - total costs.

Hence, the profit polynomial is gotten by;

`\begin{gathered} P=R-C \\ P=(-28x^2+360x+29000)-(300x+1100) \\ P=-28x^2+360x+29000-300x-1100 \\ P=-28x^2+360x-300x+29000-1100 \\ P=-28x^2+60x+27900 \end{gathered}`

Hence, the profit polynomial is;

`P=-28x^2+60x+27900`

The profits earned if 40 units are rented is;

`\begin{gathered} x=40 \\ \\ P=-28x^2+60x+27900 \\ P=-28(40^2)+60(40)+27900 \\ P=-28(1600)+2400+27900 \\ P=-44800+30300 \\ P=-14500 \\ \text{But the model can not be negative,} \\ \\ \text{Hence, the profit is 14500} \end{gathered}`

Therefore, the profit earned on 40 rented apartments is 14,500