If the demand function for a commodity is given by the equation p2 + 12q = 1800 and the supply function is given by the equation 900 − p2 + 6q = 0,
find the equilibrium quantity and equilibrium price.
I will assume your two equations are
p^2 + 12q = 1800 and 900 − p^2 + 6q = 0
p^2 + 12q = 1800
− p^2 + 6q = -900
add them:
18q = 900
q = 50
plug into p^2 + 12q = 1800 and use the positive answer to p
To find the equilibrium quantity and price, we need to find the values of q and p that satisfy both the demand and supply equations simultaneously.
1. Start by setting the demand and supply equations equal to each other:
p^2 + 12q = 1800
900 − p^2 + 6q = 0
2. Rearrange the supply equation to isolate p:
p^2 - 6q = 900
3. Substitute this expression for p^2 in the demand equation:
(p^2 - 6q) + 12q = 1800
4. Simplify the equation:
p^2 + 6q = 1800
5. Combine like terms:
2p^2 + 12q = 3600
6. Divide the equation by 2 to simplify it:
p^2 + 6q = 1800
7. Now we have a system of two equations:
p^2 + 6q = 1800
p^2 - 6q = 900
8. Add the two equations together to eliminate q:
2p^2 = 2700
9. Solve for p:
p^2 = 1350
p = ± √1350
p ≈ ± 36.74
10. Since price cannot be negative, we take the positive value:
p ≈ 36.74
11. Substitute this value of p back into one of the original equations to find q. Let's use the demand equation:
(36.74)^2 + 12q = 1800
1349.7476 + 12q = 1800
12q = 450.2524
q ≈ 37.52
The equilibrium quantity is approximately 37.52 units and the equilibrium price is approximately $36.74.
To find the equilibrium quantity and equilibrium price, we need to solve the system of equations consisting of the demand and supply functions.
The demand function is given by p^2 + 12q = 1800, and the supply function is given by 900 - p^2 + 6q = 0.
We can solve these equations simultaneously by substitution or elimination method.
Let's use the substitution method:
1. Solve the supply equation for p^2 in terms of q: p^2 = 6q - 900.
2. Substitute this expression for p^2 in the demand equation: (6q - 900) + 12q = 1800.
3. Simplify the equation: 18q - 900 = 1800.
4. Move -900 to the other side: 18q = 1800 + 900 = 2700.
5. Divide both sides by 18: q = 2700 / 18 = 150.
We have found the equilibrium quantity, which is q = 150.
Now, substitute this value of q back into either the demand or supply equation to find the equilibrium price.
Using the supply equation: 900 - p^2 + 6q = 0.
Substituting q = 150: 900 - p^2 + 6(150) = 0.
Simplifying: 900 - p^2 + 900 = 0.
Combining like terms: -p^2 + 1800 = 0.
Rearranging: -p^2 = -1800.
Dividing both sides by -1: p^2 = 1800.
Taking the square root of both sides: p = ±√1800.
The ± sign indicates that there are two possible equilibrium prices, one positive and one negative. However, since prices cannot be negative, we only consider the positive value.
Calculating positive value: p = √1800 ≈ 42.43.
Therefore, the equilibrium price is approximately p = 42.43.
In conclusion, the equilibrium quantity is q = 150, and the equilibrium price is approximately p = 42.43.