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.

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

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

Using the slope equation we have:

[tex]m=\frac{50-47}{780-825}=\frac{-3}{45}=\frac{-1}{15}[/tex]

Thus the linear equation is:

[tex]x=\frac{-1}{15}p+p_{0}[/tex]

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

[tex]50=\frac{-1}{15}780+p_{0} \\\\50=-52+p_{0} \\\\p_{0}=102 \\[/tex]

Finally, the linear equation is:

[tex]x=\frac{-1}{15}p+102[/tex]

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:

[tex]x=\frac{-1}{15}p+102[/tex]

where [tex]p=$795[/tex], then:

[tex]x=\frac{-1}{15}795+102\\ \\x=-53+102\\\\x=49[/tex]

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