Answer :
Sure! Let's solve the problem step by step.
Step 1: Defining the Revenue and Cost Equations
The revenue generated by renting out the apartments is given by the polynomial:
[tex]\[ R(x) = -11x^2 + 380x + 29000 \][/tex]
where [tex]\( x \)[/tex] is the number of apartments rented.
The monthly cost to maintain the apartments is represented by:
[tex]\[ C(x) = 120x + 1880 \][/tex]
Step 2: Calculating the Profit Polynomial
Profit is the difference between revenue and cost. So, the profit polynomial [tex]\( P(x) \)[/tex] can be found using the formula:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given expressions for [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex]:
[tex]\[ P(x) = (-11x^2 + 380x + 29000) - (120x + 1880) \][/tex]
Simplify the expression:
1. Combine the linear terms: [tex]\( 380x - 120x = 260x \)[/tex]
2. Combine the constant terms: [tex]\( 29000 - 1880 = 27120 \)[/tex]
Thus, the profit polynomial is:
[tex]\[ P(x) = -11x^2 + 260x + 27120 \][/tex]
Step 3: Calculating the Profit for 59 Units Rented
To find the profit when 59 units are rented, substitute [tex]\( x = 59 \)[/tex] into the profit polynomial:
[tex]\[ P(59) = -11(59)^2 + 260(59) + 27120 \][/tex]
After substituting these values and simplifying, the profit earned when 59 units are rented is found to be 4169.
Therefore, the profit for renting 59 units is [tex]\(\boxed{4169}\)[/tex].
Step 1: Defining the Revenue and Cost Equations
The revenue generated by renting out the apartments is given by the polynomial:
[tex]\[ R(x) = -11x^2 + 380x + 29000 \][/tex]
where [tex]\( x \)[/tex] is the number of apartments rented.
The monthly cost to maintain the apartments is represented by:
[tex]\[ C(x) = 120x + 1880 \][/tex]
Step 2: Calculating the Profit Polynomial
Profit is the difference between revenue and cost. So, the profit polynomial [tex]\( P(x) \)[/tex] can be found using the formula:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given expressions for [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex]:
[tex]\[ P(x) = (-11x^2 + 380x + 29000) - (120x + 1880) \][/tex]
Simplify the expression:
1. Combine the linear terms: [tex]\( 380x - 120x = 260x \)[/tex]
2. Combine the constant terms: [tex]\( 29000 - 1880 = 27120 \)[/tex]
Thus, the profit polynomial is:
[tex]\[ P(x) = -11x^2 + 260x + 27120 \][/tex]
Step 3: Calculating the Profit for 59 Units Rented
To find the profit when 59 units are rented, substitute [tex]\( x = 59 \)[/tex] into the profit polynomial:
[tex]\[ P(59) = -11(59)^2 + 260(59) + 27120 \][/tex]
After substituting these values and simplifying, the profit earned when 59 units are rented is found to be 4169.
Therefore, the profit for renting 59 units is [tex]\(\boxed{4169}\)[/tex].
