Find the number of mattresses demanded per month if the unit price is set at $200/mattress. (Round your answer to the nearest integer.)

Question

Consumers' Surplus, The demand function for a certain brand of mattress is given by

p = 550e^−0.09x

where p is the unit price in dollars and x (in units of a hundred) is the quantity demanded per month.

(a) Find the number of mattresses demanded per month if the unit price is set at $200/mattress. (Round your answer to the nearest integer.)

 _______ mattresses

(b) Use the results of part (a) to find the consumers' surplus if the selling price is set at $200 per mattress. (Round your answer to the nearest dollar.)

$ _______

Answer 1

a) We simply need to plug in $200 for p and solve.

200 = 550e^-0.09x

0.3636 = e^-0.09x

ln(0.3636) = -0.09x

x = ln(0.3636) / -0.09 = 11.24, rounds to 11 mattresses.

b) The consumer surplus is the area under the demand curve but above the horizontal selling price line - in this case, p=$200. When the price is $200, 11 mattresses will sell (from part A). So we can set up the equation like so:

Consumer surplus = (Total area under demand curve from x=0 to 11) - (area under horizontal selling price line, p = 200, from x=0 to 11...so, a rectangle with height 200 and length 11)

= ∫₀¹¹550e^−0.09x - 200(11-0)

= 3840.36 - 2200

= 1640.36, rounds to $1640