suppose the supply and demand for a certain videotape are given by:
Supply:p=1/5q^2
Demand p=-1/5q^2+30
where p is price and q is quantity
find the equilibrium price?
wouldn't you set demand = supply, and solve for q?
To find the equilibrium price, we need to find the price at which the quantity supplied equals the quantity demanded.
Given that the supply is given by p = 1/5q^2 and the demand is given by p = -1/5q^2 + 30, we can set these two equations equal to each other:
1/5q^2 = -1/5q^2 + 30
Now, let's solve this equation step by step:
1/5q^2 + 1/5q^2 = 30
2/5q^2 = 30
q^2 = (30 * 5) / 2
q^2 = 75
Taking the square root of both sides:
q = √75
q = √(25 * 3)
q = 5√3
Now that we have the quantity, we can substitute it back into either the demand or supply equation to find the equilibrium price. Let's use the demand equation:
p = -1/5q^2 + 30
p = -1/5(5√3)^2 + 30
p = -1/5(25 * 3) + 30
p = -5 + 30
p = 25
Therefore, the equilibrium price is 25.
To find the equilibrium price, we need to find the price at which the quantity supplied equals the quantity demanded. In other words, we need to find the price that makes the supply and demand equations equal to each other.
The supply equation is given as:
Supply: p = (1/5)q^2
The demand equation is given as:
Demand: p = (-1/5)q^2 + 30
To find the equilibrium price, we can set the two equations equal to each other:
(1/5)q^2 = (-1/5)q^2 + 30
To simplify this equation, let's first combine the similar terms on both sides:
(1/5)q^2 + (1/5)q^2 = 30
Now, let's add the two fractions on the left side:
(2/5)q^2 = 30
To isolate q^2, we can multiply both sides by 5/2:
q^2 = (30)(5/2)
Simplifying the right side:
q^2 = 75
To solve for q, we take the square root of both sides:
q = sqrt(75)
Taking the square root, you get:
q = 8.66
Now that we have the value of q, we can substitute it back into either the supply or demand equation to find the equilibrium price.
Using the supply equation:
p = (1/5)(8.66)^2
p = 14.51
Using the demand equation:
p = (-1/5)(8.66)^2 + 30
p = 14.51
Hence, the equilibrium price is approximately $14.51.