If the supply function for a company is p = q^2 + 8q +16 and the demand function is p = -3q^2 + 6q + 36. Find the equilibrium quantity and equilibrium price.
p = p so
q^2 + 8 q + 16 = -3q^2 + 6q + 36
4 q^2 + 2 q - 20 = 0
2 q^2 + q - 10 = 0
(2 q + 5 ) (q - 2) = 0
so q = 2 is the positive root
now go back and find p
To find the equilibrium quantity and price, we need to set the supply function equal to the demand function and solve for q.
Setting p (price) in both equations equal to each other, we have:
q^2 + 8q + 16 = -3q^2 + 6q + 36
Rearranging the equation:
4q^2 - 2q - 20 = 0
Dividing through by 2:
2q^2 - q - 10 = 0
To solve this quadratic equation, we can use the quadratic formula:
q = (-b ± √(b^2 - 4ac))/(2a)
For our equation, a = 2, b = -1, and c = -10. Plugging these values into the formula, we get:
q = (-(-1) ± √((-1)^2 - 4(2)(-10)))/(2(2))
q = (1 ± √(1 + 80))/4
q = (1 ± √81)/4
q = (1 ± 9)/4
This gives us two possibilities:
q1 = (1 + 9)/4 = 10/4 = 2.5
q2 = (1 - 9)/4 = -8/4 = -2
Since a negative quantity doesn't make sense in this context, we discard q2 = -2.
Therefore, the equilibrium quantity is q = 2.5.
To find the equilibrium price, we can substitute this quantity into either the supply or demand function. We'll use the supply function:
p = q^2 + 8q + 16
p = (2.5)^2 + 8(2.5) + 16
p = 6.25 + 20 + 16
p = 42.25
So, the equilibrium quantity is 2.5 units and the equilibrium price is $42.25.
To find the equilibrium quantity and equilibrium price, we need to set the supply function equal to the demand function and solve for q and p.
1. Set the supply function equal to the demand function:
q^2 + 8q + 16 = -3q^2 + 6q + 36
2. Rearrange the equation to standard form:
4q^2 - 2q - 20 = 0
3. Solve this quadratic equation using the quadratic formula:
The quadratic formula is given by:
q = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 4, b = -2, and c = -20.
q = (-(-2) ± √((-2)^2 - 4(4)(-20))) / (2*4)
= (2 ± √(4 + 320)) / 8
= (2 ± √324) / 8
= (2 ± 18) / 8
So we have two possible values for q:
q1 = (2 + 18) / 8 = 20 / 8 = 2.5
q2 = (2 - 18) / 8 = -16 / 8 = -2
4. Substitute the values of q back into the supply or demand function to find the equilibrium price, p.
Using the supply function:
p = (2.5)^2 + 8(2.5) + 16
= 6.25 + 20 + 16
= 42.25
So the equilibrium price is $42.25 when the quantity is 2.5.
Similarly, using the demand function:
p = -3(2.5)^2 + 6(2.5) + 36
= -18.75 + 15 + 36
= 32.25
So the equilibrium price is also $32.25 when the quantity is 2.5.
Therefore, the equilibrium quantity is 2.5 and the equilibrium price is $32.25 or $42.25, depending on which function is used to calculate the equilibrium price.