Assume that the demand function for a certain commodity had the following form where x is the quantity demanded, measured in units

demand-functions

#1

Assume that the demand function for a certain commodity had the following form where x is the quantity demanded, measured in units of a thousand and p is the unit price in dollars.
p = sqrt(-ax^2 + b) text( where ) (a>=0,b>=0)
Suppose the quantity demanded is 6000 (x = 6) when the unit price is $8.00 and 8000 (x = 8) when the unit price is $6.00. Determine the demand equation

Answer:

Given x= 6 when P= 8
also x= 8 when P= 6
Substituting these in given demand- price equation we get
8 = sqrt(- 36a +b)
6 = sqrt(- 64a +b)
Taking square on both sides we get
64 = -36a + b -(1)
36 = -64a + b -(2)
Subtracting eq(2) from eq(1), we get
28 = 28a
a=1
Substituting this in equation 1, we get
b= 100
Therefore
P=sqrt(-x^2+100)
P^2= -x^2 + 100
x = sqrt(100 - P^2)
This is the required demand equation.