The consumer demand equation for tissues is given by q = (102 − p)2

Question

The consumer demand equation for tissues is given by q = (102 − p)²,

 where p is the price per case of tissues and q is the demand in weekly sales.

(a) Determine the price elasticity of demand E when the price is set at $32. (Round your answer to three decimal places.)

E = ____

Interpret your answer.

The demand is going  ? up down by  % per 1% increase in price at that price level.

(b) At what price should tissues be sold to maximize the revenue? (Round your answer to the nearest cent.)

$ ____

(c) Approximately how many cases of tissues would be demanded at that price? (Round your answer to the nearest whole number.)

___ cases per week

Answer 1

q = (102 -p)^2 =102^2 -204p +p^2

q' = dq/dp = 2(102-p)(-1) = -204 +2p

q'(32) = -204 +2(32) = -204 + 64 = -140

a $1 increase in price will cause 140 decrease in quantity demanded

q(32) =(102-32)^2 = 70^2 = 4900

q(33) = 102-33)^2 = 69^2 = 4761

4900-4761 = 139

q(31) = 102-31)^2 = 71^2 = 5041

5041-4900 = 141

(139+141)/2 = 140

$2 increase from $31 to $33 causes a decrease in quantity purchased of 280

5041-4761 = 280

280/2 = 140 decrease per $1 increase in price

price elasticity of demand = % change in quantity divided by % change in price

= change in quantity divided by quantity all over change in price divided by price

= (change in quantity times price) divided by change in price times quantity)

using the derivative gives the instantaneous price elasticity at a point on the demand curve where "change in quantity" and "change in price" approach zero.

a 1% increase in price when P=$32 is .01(32) = a 32 cent increase to $32.32

q = (102-p)^2

q(32) = 70^2 = 4900

q(32.32) = (102-32.32)^2 = 69.68^2 =4855.3024

that's a decrease in quantity demanded of 4900-4855.3024 = 44.6976

44.6976/4900 = .00912 = 0.912% decrease in quantity demanded = almost -1%

IF you took the demand function literally (which is not realistic), the maximum revenue would be when price approached infinity dollars. But that's nonsensical.

at P =$0, revenue = $0, when P=$102, revenue = $0

profit maximizing price is somewhere between 0 and $102

1/3 the way between them is 102/3 = 34

$34.00 is the revenue maximizing price, not just rounded off to nearest cent, but exactly $34,00

R(33) = 33(102-33)^2 = 33(69)^2 = 33(44761) = 147,113

R(33.99) = 33.99(102-33.99)^2 = 33.99(68.01)^2 = 157215.9898

R(34) = 34(102-34)^2 = 34(68)^2 = 34(4624) = 157,216

R(34.01) = 34.01(102-34.01)^2 = 157,215.9898

R(35) = 35(102-35)^2 = 35(67)^2 = 35(4489) = 157,115

q= 4624 when P=$34

Revenue = pq = p(102-p)^2 = 102^2(p) -204p^2 + p^3

take the derivative and set = 0

R' =102^2 - 408p +3p^2 = 0

p^2 -136p + 10404/3 = 0

p = 136/2 + or - (1/2)sqr(136^2 -4(10404)/3)

= 68 + or - (1/2)sqr(18,496-13872)

=68+or -34

=$34 or $102

$34 maximizes revenue, $102 minimizes revenue