For the following demand function, find the elasticity of demand, and the values of q (if any) at which total revenue is maximized.
q=37,700-7p^2
To find the elasticity of demand, we would need to differentiate the demand function with respect to the price (p) and then multiply it by p/Q where Q is the quantity demanded at price p.
q = 37,700 - 7p^2
First, differentiate q with respect to p:
dq/dp = 0 - 14p
Now, let's find the quantity demanded at price p:
q = 37,700 - 7p^2
q = 37,700 - 7(5500)^2
q = 37,700 - 7(30,250,000)
q = 37,700 - 211,750,000
q = -211,712,300
Now, we can find the elasticity of demand:
Elasticity of demand = (dq/dp) * (p/Q)
Elasticity of demand = (-14p) * (p/ -211,712,300)
Elasticity of demand = 14p^2 / 211,712,300
To find the quantity at which total revenue is maximized, we need to find the critical points of the total revenue function. Total revenue is given by the product of price and quantity demanded:
TR = p*q
TR = p(37,700 - 7p^2)
TR = 37,700p - 7p^3
To maximize total revenue, we need to find the critical points of the total revenue function:
dTR/dp = 37,700 - 21p^2
Set dTR/dp = 0 to find the critical points:
37,700 - 21p^2 = 0
21p^2 = 37,700
p^2 = 37,700 / 21
p^2 = 1795.24
p = sqrt(1795.24)
p = 42.38 (approx.)
Therefore, the optimal price for maximizing total revenue is approximately 42.38. To find the corresponding quantity, plug this price back into the demand function:
q = 37,700 - 7(42.38)^2
q = 37,700 - 7(1800.71)
q = 37,700 - 12,606.97
q = 25,093.03
Therefore, the optimal quantity for maximizing total revenue is approximately 25,093.03.