Find the equilibrium quantity and equilibrium price for the commodity whose supply and demand functions are given.
Supply: [tex]p=q^2+30 q[/tex]
Demand: [tex]p=-2 q^2+10 q+12,000[/tex]
(On this one, just give me the EQUILIBRIUM QUANTITY.)
Answer (1)
Set the supply and demand functions equal: q 2 + 30 q = − 2 q 2 + 10 q + 12000 .
Rearrange the equation into standard quadratic form: 3 q 2 + 20 q − 12000 = 0 .
Solve for q using the quadratic formula: q = 2 ( 3 ) − 20 ± 2 0 2 − 4 ( 3 ) ( − 12000 ) .
Discard the negative solution, so the equilibrium quantity is: 60 .
Explanation
Problem Analysis We are given the supply and demand functions for a commodity and asked to find the equilibrium quantity. The equilibrium quantity is the quantity at which the supply and demand functions are equal.
Setting Supply Equal to Demand To find the equilibrium quantity, we set the supply and demand functions equal to each other: q 2 + 30 q = − 2 q 2 + 10 q + 12000
Rearranging the Equation Now, we rearrange the equation to get a quadratic equation in standard form: 3 q 2 + 20 q − 12000 = 0
Applying the Quadratic Formula We can solve this quadratic equation using the quadratic formula: q = 2 a − b ± b 2 − 4 a c where a = 3 , b = 20 , and c = − 12000 .
Calculating the Roots Plugging in the values, we get: q = 2 ( 3 ) − 20 ± 2 0 2 − 4 ( 3 ) ( − 12000 ) q = 6 − 20 ± 400 + 144000 q = 6 − 20 ± 144400 q = 6 − 20 ± 380
Finding the Two Possible Solutions This gives us two possible solutions for q :
q 1 = 6 − 20 + 380 = 6 360 = 60 q 2 = 6 − 20 − 380 = 6 − 400 = − 3 200 ≈ − 66.67
Determining the Equilibrium Quantity Since quantity cannot be negative, we discard the negative solution. Therefore, the equilibrium quantity is: q = 60
Examples
Understanding equilibrium quantity is crucial in business. For instance, if you're selling lemonade, knowing the equilibrium quantity helps you determine how much lemonade to prepare each day. If you prepare too much, you'll have waste; if you prepare too little, you'll miss out on potential sales. By analyzing supply and demand, you can optimize your production to meet customer needs without incurring unnecessary costs.
