Question

You have been hired as a marketing consultant to Johannesburg Burger Supply, Inc., and you wish to come up with a unit price for its hamburgers in order to maximize its weekly revenue. To make life as simple as possible, you assume that the demand equation for Johannesburg hamburgers has the linear form q = mp + b, where p is the price per hamburger, q is the demand in weekly sales, and m and b are certain constants you must determine.

Your market studies reveal the following sales figures: When the price is set at $2.00 per hamburger, the sales amount to 9000 per week, but when the price is set at $4.00 per hamburger, the sales drop to zero.
(A) Use these data to find the linear demand function q(p), where p is the price per hamburger and q is the number of hamburgers they sell at that price per week.

Answer 1

The demabd function is:

`q(p)= -4500p+18000`

Explanation:

The demand follow the linear equation
`q(p)=mp+b`

1. When p=$2.00 and q=9000, the equation is:

`9000=m(2.00)+b (1)`

2. Whe p=$4.00 and q=0, the equation is:

`0=m(4.00)+b (2)`

3. Solve equation (1) for b:

`9000=2m+b\\9000-2m=2m+2m+b\\9000-2m=b`

4. Replace the value of b in equation (2)

`0=4m+b\\0=4m+9000-2m\\0=2m+9000\\0-9000=2m+9000-9000\\-9000=2m\\(-9000)/(2m)=(2m)/(2m)\\ -4500=m`

The value of m is 4500

5. Calculate b, replacing m:

`0=4m+b\\\\0=4(-4500)+b\\0=-18000+b\\18000=b`

The value of b is 18000