Question

The demand function for our product is: Q=90-0.1P^2

a.) Fine the elasticity when Q=50.
b.) Is the demand elastic or inelastic?
If we raise the price will our total revenue increase or will it decrease? Find the maximum revenue and the values of P and Q which will maximize the revenue.

Answer 1

Assuming that "elasticity" = P:

A.
`Q=90-0.1P^2\\50=90-0.1P^2\\-40=-0.1P^2\\400=P^2\\20=P\\\\(dQ)/(dP)=-0.2P\\E=(P)/(Q)*(dQ)/(dP)\\E=((20))/((90))(-0.2P)\\E=(-2(20))/(45)\\E=(-8)/(9)` - Elasticity = -0.889

B.
`0>(-8)/(9)>-1>-\infty` - The demand is inelastic because the elasticity > -1.

(C). Set P and Q to 1 in two separate functions. If Q < P revenue will increase. If Q > P revenue will decrease.


`Q=90-0.1P^2\\Q=90-0.1(1)^2\\Q=90-0.1\\Q=89.9\\\\(1)=90-0.1P^2\\-89=-0.1P^2\\890=P^2\\√(890)=P\\P=29.833\\Q>P\\(89.9)>(29.833)`

Q > P therefore revenue will decrease.

(D).
`Q=90-0.1P^2\\(dQ)/(dP)=-0.2P\\-0.2P=0\\P=0`

One obviously won't be able to maximize revenue if their price per unit, P, equals 0. Quantity of a product can only be sold in whole, so the closest integer to 90 is 89. The value of P that maximizes revenue is
`Q=90-0.1P^2\\(89)=90-0.1P^2\\-1=-0.1P^2\\10=P^2\\√(10)=P` - sqrt(10).

Therefore, the values of P and Q that maximize revenue are 3.162 and 89, respectively.