If the demand function for a commodity is given by the equation
p2 + 16q = 1800
and the supply function is given by the equation
500 − p2 + 2q = 0,
find the equilibrium quantity and equilibrium price.
To find the equilibrium quantity and equilibrium price, we need to solve the system of equations formed by the demand and supply functions.
The demand function is given by:
p^2 + 16q = 1800 ----(1)
The supply function is given by:
500 - p^2 + 2q = 0 ----(2)
To solve the system of equations, we can use the substitution method.
First, let's solve equation (2) for p^2:
p^2 = 500 + 2q
Now we substitute this value of p^2 in equation (1):
(500 + 2q) + 16q = 1800
500 + 18q = 1800
18q = 1800 - 500
18q = 1300
q = 1300 / 18
q ≈ 72.22
Substitute this value of q into equation (2):
500 - p^2 + 2(72.22) = 0
500 - p^2 + 144.44 = 0
-p^2 = -644.44
p^2 = 644.44
p ≈ √644.44
p ≈ 25.4
Therefore, the equilibrium quantity is approximately 72.22 units and the equilibrium price is approximately $25.4.