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

p = −0.01x2 − 0.2x + 12
where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the market price is set at $9/disc. (Round your answer to two decimal places.)

To determine the consumer's surplus, we first need to find the equilibrium quantity where the market price equals the wholesale price. In this case, the market price is set at $9/disc.

Let's start by substituting the market price, $9, into the demand function:

9 = -0.01x^2 - 0.2x + 12

Now we can solve this quadratic equation to find the equilibrium quantity (x). Rearrange the equation to match the quadratic form (ax^2 + bx + c = 0), and then we can apply the quadratic formula:

0.01x^2 + 0.2x - 3 = 0

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 0.01, b = 0.2, and c = -3:

x = (-0.2 ± √(0.2^2 - 4 * 0.01 * -3)) / (2 * 0.01)

Simplifying the expression further:

x = (-0.2 ± √(0.04 + 0.12)) / 0.02

x = (-0.2 ± √0.16) / 0.02

x = (-0.2 ± 0.4) / 0.02

Now we have two possible solutions for x:

x₁ = (-0.2 + 0.4) / 0.02
x₁ = 20

x₂ = (-0.2 - 0.4) / 0.02
x₂ = -30

Since we're talking about the quantity demanded each week, x cannot be negative. So we discard x₂ = -30 and consider only x₁ = 20 as the equilibrium quantity.

Now that we know the equilibrium quantity is 20 (in units of a thousand), we can calculate the consumers' surplus.

Consumers' surplus represents the difference between the maximum price consumers are willing to pay and the actual price they pay. In this case, the maximum price consumers are willing to pay can be determined by substituting the equilibrium quantity (20) into the demand function:

p = -0.01x^2 - 0.2x + 12
p = -0.01(20)^2 - 0.2(20) + 12
p ≈ $9.6 (rounded to two decimal places)

The actual price is given as $9/disc.

The consumers' surplus can be calculated as:

Consumers' Surplus = (Maximum Price Consumers are Willing to Pay) - (Actual Price)

Consumers' Surplus = $9.6 - $9 = $0.60 (rounded to two decimal places)

Therefore, the consumers' surplus for the given market price is approximately $0.60.