Determine the consumers' surplus (in dollars) if the market price is set at $9/disc. (Round your answer to two decimal places.)

Question

Consumers' Surplus

 The demand function for a certain brand of CD is given by

p = −0.01x² − 0.3x + 13

where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. 

Answer 1

Demand function:

p = −0.01x² − 0.3x + 13

Market price:

p= 9

Plugin 9 to the demand function:

9= −0.01x² − 0.3x + 13

0 = −0.01x² − 0.3x + 4

multiply -100 on both sides of the equation:

0 =x² +30x - 400

0 =(x+40)(x-10)

x₁ = -40, x₂ = 10

-40 is not feasible. Therefore the x=10.

Let's set the integral:

∫₀¹⁰ (−0.01x² − 0.3x + 4)dx

=[-(1/300)x³-0.15x² +4x]₀¹⁰

=-(1/300)(10)³ - 0.15(10)² +4(10)

= -10/3 -15 + 40 =65/3 ≈ $21.67