Question

The demand function for a certain product, p=D(q), is linear, where p is the price per item in dollars and q is the quantity demanded. If p increases by 8$, market research shows that q drops by two items. In addition, 50 items are purchased if the price is 250. Find a formula for q as a function

Answer 1

`q=-(1)/(4)p+112.5`

Explanation:

Given:

`p=D(q)` is a linear model for demand function
`D`

where
`p` represents the price per item in dollars and
`q` is the quantity demanded.

With increase in
`p` by $8 there is a decrease in
`q` by 2 items.

For
`q=50` items purchased the price
`p=250`.

To find
`q` as a function of
`p` which is
`q=F(p)`

Since its a linear model, so we can find rate of change or slope of line from the given data.

Slope
`(m)=\frac{\textrm{Change in q}}{\textrm{Change in p}} = (-2)/(8)=(-1)/(4)`

So, the function can be written as:

`q=-(1)/(4)p+b`

where
`b` is the initial value or the y-intercept.

Using the point
`(250,50)`

`50=-(1)/(4)(250)+b`

`50=-62.5+b`

Adding 62.5 both sides.

`50+62.5=-62.5+b+62.5`

∴
`b=112.5`

So,
`q=-(1)/(4)p+112.5`