Q:

A real estate firm owns the Kasumba Garden Apartments which consists of 90 apartments. At $350 per month every apartment can be rented. However, for each $10 per month increase there will be two vacancies with no possibility of filling them. The firm wants to receive $31 980 per month from rents. What rent should be charged for each apartment?

Accepted Solution

A:
Answer:The rent should be $ 390 or $ 410.Step-by-step explanation:Given,The original monthly rent of an apartment = $350,Also, the original number of apartment that could be filled = 90,Let the rent is increased by x times of $ 10,That is, the new monthly rent of an apartment =( 350 + 10x ) dollarsSince, for each $10 per month increase there will be two vacancies with no possibility of filling them. Thus, the new number of filled apartments = 90 - 2x,Hence, the total revenue of the firm = ( 90 - 2x )(350 + 10x ) dollars,According to the question,( 90 - 2x )(350 + 10x ) = 31,980 [tex]90(350)+90(10x)-2x(350)-2x(10x)=31980[/tex][tex]31500+900x-700x-20x^2=31980[/tex][tex]-20x^2+200x+31500-31980=0[/tex][tex]20x^2-200x+480=0[/tex]By the quadratic formula,[tex]x=\frac{200\pm \sqrt{(-200)^2-4\times 20\times 480}}{40}[/tex][tex]x=\frac{200\pm \sqrt{1600}}{40}[/tex][tex]x=\frac{200\pm 40}{40}[/tex][tex]\implies x=\frac{200+40}{40}\text{ or }x=\frac{200-40}{40}[/tex][tex]\implies x=6\text{ or } x =4[/tex]Hence, the new rent of each apartment, if x = 6, is $ 410,While, if x = 4, is $ 390