Answer :
Answer:
[tex]q=-\frac{1}{4}p+112.5[/tex]
Step-by-step explanation:
Given:
[tex]p=D(q)[/tex] is a linear model for demand function [tex]D[/tex]
where [tex]p[/tex] represents the price per item in dollars and [tex]q[/tex] is the quantity demanded.
With increase in [tex]p[/tex] by $8 there is a decrease in [tex]q[/tex] by 2 items.
For [tex]q=50[/tex] items purchased the price [tex]p=250[/tex].
To find [tex]q[/tex] as a function of [tex]p[/tex] which is [tex]q=F(p)[/tex]
Since its a linear model, so we can find rate of change or slope of line from the given data.
Slope[tex](m)=\frac{\textrm{Change in q}}{\textrm{Change in p}} = \frac{-2}{8}=\frac{-1}{4}[/tex]
So, the function can be written as:
[tex]q=-\frac{1}{4}p+b[/tex]
where [tex]b[/tex] is the initial value or the y-intercept.
Using the point [tex](250,50)[/tex]
[tex]50=-\frac{1}{4}(250)+b[/tex]
[tex]50=-62.5+b[/tex]
Adding 62.5 both sides.
[tex]50+62.5=-62.5+b+62.5[/tex]
∴ [tex]b=112.5[/tex]
So, [tex]q=-\frac{1}{4}p+112.5[/tex]