Elasticity question

Question

The Physics Club sells E = mc² T-shirts at a local flea market. Unfortunately, the club's previous administration has been losing money for years, so you decide to do an analysis of the sales. A quadratic regression based on old sales data reveals the following demand equation for the T-shirts.

q = -2p² + 30p. (9 < p < 15)

(a) Obtain a formula for the price elasticity of demand for E = mc² T-shirts.

E= __________

(b) Compute the elasticity of demand if the price is set at $9 per shirt. (Round your answer to two decimal places.)

__________

Interpret the result.

The demand for E = mc² T-shirts is going ___?___ by about ______% per 1% increase in the price.

(c) How much should the Physics Club charge for the T-shirts to obtain the maximum daulily revenue?

$______

What will the revenue be? $______

Answer 1

The formula for the elasticity function is

(p/q) * (dp/dq)

where

p - price

q- demand function

(dp/dq) is the derivative of the demand function with respect to price.

a. [p/q] * [dp/dq] =[ p/(-2p² + 30p) ] * [ -4p+30 ] = [1/(-2p+30)] * [ -4p+30 ] = (-4p+30/-2p+30) = (-1/-1) ((4p-30)/ (2p-30)) = (4p-30)/ (2p-30)

b. (4(9)-30)/(2(9)-30)=6/-12=-1/2=-0.5

interpretation: down by .5%

c. Since the leading coefficient of the demand function q is negative, we know it has a maximum at its vertex where p=-b/2a=-30/-4=7.50. The unconstrained price associated with maximum revenue is $7.50, so they should charge $9.00 per shirt. This is because, in your problem, price is constrained to be (9 < p < 15).

The maximum revenue at the constrained optimal price is -2($9.00^2)+30($9.00)=$108.00