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

A) The linear equation is
`x=(-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.

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=(50-47)/(780-825)=(-3)/(45)=(-1)/(15)`

Thus the linear equation is:

`x=(-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=(-1)/(15)780+p_(0)  \\\\50=-52+p_(0)  \\\\p_(0)=102  \\`

Finally, the linear equation is:

`x=(-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=(-1)/(15)p+102`

where
`p=$795`, then:

`x=(-1)/(15)795+102\\ \\x=-53+102\\\\x=49`

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