Question

A real estate office manages an apartment complex with 50 units. When the rent is $780 per month, all 50 units are occupied. However, when the rent is $825, the average number of occupied units drops to 47. Assume that the relationship between the monthly rent p and the demand x is linear (Note:The term demand refers to thenumber of occupied units.)
(a) Write a linear equation giving the demand x in terms of the rent p. (b) Linear extrapolation - Use a graphing utility to graph the demand equation and use the trace feature to predict the
number of units occupied when the rent is raised to $855. (c) Linear interpolation - Predict the number of units occupied when the rent is lowered to $795.

Answer 1

Answer:

A) The linear equation is $x=\frac{-1}{15}p+102$

B) When the rent is raised to $855 the number of units occupied is 45.

C) When the rent is lowered to $795 the number of units occupied is 49.  

Step-by-step explanation:

A) A linear equation for the demand is written as $x=mp+p_{0}$, where $m$ is the slope, $x$ is the number of occupied units, $p$ is the rent.

$m$ is calculated using the problem information. When the rent is $p= 780$ then $x=50$ and when the rent is $p= 825$ then $x=47$.

Using the slope equation we have:

$m=\frac{50-47}{780-825}=\frac{-3}{45}=\frac{-1}{15}$

Thus the linear equation is:

$x=\frac{-1}{15}p+p_{0}$

In order to calculate $p_{0}$ we use the problem information, When the rent is $p= 780$ then number of occupied units is $x=50$, thus:

$50=\frac{-1}{15}780+p_{0} \\\\50=-52+p_{0} \\\\p_{0}=102$

Finally, the linear equation is:

$x=\frac{-1}{15}p+102$

B) The demand equation is plot in the attached file, the number of units occupied when the rent is raised to $855 is 45.

C) In order to predict the number of occupied units lets use the equation:

$x=\frac{-1}{15}p+102$

where $p= 795$, then:

$x=\frac{-1}{15}795+102\\ \\x=-53+102\\\\x=49$

Thus, when the rent is lowered to $795 the number of units occupied is 49.