If the demand function for a commodity is given by the equation p2 + 16q = 1200 and the supply function is given by the equation 300 − p2 + 6q = 0, find the equilibrium quantity and equilibrium price.
To find the equilibrium quantity and price, we need to solve the demand and supply equations simultaneously.
First, let's express the demand and supply equations in terms of q:
Demand: p^2 + 16q = 1200 ...(1)
Supply: 300 - p^2 + 6q = 0 ...(2)
To solve this system of equations, we can rearrange equation (2) to express p^2 in terms of q:
p^2 = 300 + 6q.
Now we substitute this expression for p^2 in equation (1):
(300 + 6q) + 16q = 1200.
Combine like terms:
22q + 300 = 1200.
Subtract 300 from both sides:
22q = 900.
Divide both sides by 22:
q = 900/22 ≈ 40.91.
Now substitute this value of q back into equation (2) to find the equilibrium price:
p^2 = 300 + 6(40.91),
p^2 = 300 + 245.45,
p^2 = 545.45.
Taking the square root of both sides:
p ≈ √545.45 ≈ 23.35.
Therefore, the equilibrium quantity is approximately 40.91 units, and the equilibrium price is approximately $23.35.
To find the equilibrium quantity and equilibrium price, we need to find the point at which the demand function and supply function intersect, that is, when the quantity demanded equals the quantity supplied.
Given the demand function: p^2 + 16q = 1200, where p represents the price and q represents the quantity demanded.
And the supply function: 300 - p^2 + 6q = 0, where p represents the price and q represents the quantity supplied.
To solve for equilibrium, we need to equate the quantity demanded and the quantity supplied:
p^2 + 16q = 300 - p^2 + 6q
Combine similar terms:
2p^2 - 10q = 300
Rearrange the equation:
2p^2 = 10q + 300
Divide both sides by 2:
p^2 = 5q + 150
Now, we have an equation in terms of either p^2 or q, we can solve for one variable and then substitute it into one of the original equations to find the other variable.
Let's solve for q in terms of p:
5q = p^2 - 150
Divide both sides by 5:
q = (p^2 - 150) / 5
Now we can substitute this expression for q into either the demand or supply equation to find the equilibrium price.
Let's use the demand function:
p^2 + 16((p^2 - 150) / 5) = 1200
Multiply both sides by 5 to remove fractions:
5p^2 + 80(p^2 - 150) = 6000
Expand and combine like terms:
5p^2 + 80p^2 - 12000 = 6000
Combine similar terms:
85p^2 - 12000 = 6000
Move the constant term to the other side:
85p^2 = 18000
Divide both sides by 85:
p^2 = 211.7647
Take the square root of both sides to solve for p (ignoring the negative square root since price cannot be negative):
p = √(211.7647)
p ≈ 14.55
Now, substitute this value of p back into either the demand or supply equation to find the equilibrium quantity (q):
Using the demand function:
q = (p^2 - 150) / 5
q = ((14.55)^2 - 150) / 5
q ≈ 17.91
Therefore, the equilibrium quantity is approximately 17.91 units and the equilibrium price is approximately $14.55.
just solve the two equations...
p^2 + 16q = 1200 ---> p^2 = 1200 - 16q
300 - p^2 + 6q = 0 ---> p^2 = 300 + 6q
300 + 6q = 1200-16q
22q = 900
q = 900/22 = 450/11 = appr 40.9
then p^2 = 300 + 6(450/11) = 4650/11 = appr 422.7