If the supply function for a commodity is given by the equation p = q2 + 12q + 700 and the demand function is given by the equation p = 1800 - 10q - q2, find the equilibrium quantity and equilibrium price. (Round your answers to two decimal places.)Thank You so much
At equilibrium,p1=p2 q2+12q+700=1800-10q-q2 2q2+22q-1100=0 q2+11q-550=0 solving q=18.59 p=18.59^2+12(18.59)+700=N1268.67
To find the equilibrium quantity and equilibrium price, we need to set the supply function equal to the demand function and solve for q (quantity).
Setting the supply function equal to the demand function:
q^2 + 12q + 700 = 1800 - 10q - q^2
Rearranging the equation and combining like terms:
2q^2 + 22q - 1100 = 0
This is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = 22, and c = -1100. We can use the quadratic formula to solve for q:
q = (-b ± sqrt(b^2 - 4ac)) / (2a)
Plugging in the values:
q = (-(22) ± sqrt((22)^2 - 4(2)(-1100))) / (2(2))
Simplifying:
q = (-22 ± sqrt(484 + 8800)) / 4
q = (-22 ± sqrt(9284)) / 4
Calculating the square root of 9284:
sqrt(9284) ≈ 96.33
Substituting this value back into the equation:
q = (-22 ± 96.33) / 4
Two possible solutions:
1. q = (-22 + 96.33) / 4 ≈ 18.08
2. q = (-22 - 96.33) / 4 ≈ -29.08
Since a negative quantity does not make sense in this context, we discard the second solution (-29.08) and focus on the positive quantity (18.08).
Now that we have the value of q, we can find the equilibrium price (p) by plugging it into either the supply or demand function.
Using the demand function (p = 1800 - 10q - q^2):
p = 1800 - 10(18.08) - (18.08)^2
p ≈ 951.82
Therefore, the equilibrium quantity is approximately 18.08 units, and the equilibrium price is approximately $951.82.