If the supply and demand functions for a commodity are given by 2p - q + 10 = 0 and (p + q)(q + 10) = 3642 respectively, find the equilibrium price and quantity that give market equilibrium. (Round your answers to two decimal places.)
Equilibrium price =
well, as I recall, the equilibrium price and quantity are where supply equals demand. So, we want
2p-q+10 = (p+q)(q+10)-3642
since q = 2p+10, that means
(p+2p+10)(2p+10+10) = 3642
6p^2+80p-3442 = 0
p = 18.195
so, q = 46.390
Check:
2(18.195)-46.390 +10 = 0
(18.195+46.390)(56.390) = 3642
The curves are at
http://www.wolframalpha.com/input/?i=plot+2p+-+q+%2B+10+%3D+0+and+%28p+%2B+q%29%28q+%2B+10%29+%3D+3642+for+p+%3D+0+to+50
To find the equilibrium price and quantity, we need to solve the system of equations formed by the supply and demand functions.
1. Supply Function: 2p - q + 10 = 0
2. Demand Function: (p + q)(q + 10) = 3642
Step 1: Solve the supply function for q in terms of p.
2p - q + 10 = 0
-q = -2p - 10
q = 2p + 10
Step 2: Substitute the solution for q in terms of p into the demand function.
(p + (2p + 10))((2p + 10) + 10) = 3642
(3p + 10)(2p + 20) = 3642
6p^2 + 80p + 200 = 3642
6p^2 + 80p - 3442 = 0
Step 3: Solve the quadratic equation for p using the quadratic formula.
p = (-b ± sqrt(b^2 - 4ac)) / 2a
p = (-80 ± sqrt(80^2 - 4 * 6 * -3442)) / (2 * 6)
Calculating the values inside the square root:
sqrt(80^2 - 4 * 6 * -3442) = sqrt(6400 + 82512) = sqrt(88912) ≈ 297.83
Substituting the values back into the quadratic formula:
p = (-80 ± 297.83) / 12
Simplifying the expression further:
p ≈ (217.83 / 12) or (377.83 / 12)
Calculating the values for p:
p ≈ 18.15 or p ≈ 31.48
Step 4: Substitute the values of p into the supply function to find the corresponding values of q.
For p ≈ 18.15:
q = 2p + 10 ≈ (2 * 18.15) + 10 ≈ 36.30 + 10 ≈ 46.30
For p ≈ 31.48:
q = 2p + 10 ≈ (2 * 31.48) + 10 ≈ 62.96 + 10 ≈ 72.96
The equilibrium price is the value of p that satisfies both the supply and demand functions. Since 18.15 and 31.48 are the possible values of p, we need to check which one satisfies both equations.
Checking for p ≈ 18.15:
Supply Function: 2(18.15) - 46.30 + 10 ≈ 36.30 - 46.30 + 10 ≈ 0 (Satisfied)
Demand Function: (18.15 + 46.30)(46.30 + 10) ≈ 64.45 * 56.30 ≈ 3633.23 (Not satisfied)
Checking for p ≈ 31.48:
Supply Function: 2(31.48) - 72.96 + 10 ≈ 62.96 - 72.96 + 10 ≈ 0 (Satisfied)
Demand Function: (31.48 + 72.96)(72.96 + 10) ≈ 104.44 * 82.96 ≈ 3645.42 (Not satisfied)
So, the equilibrium price is approximately $18.15.