The demand function for a certain brand of CD is given by the following equation where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.


P=-0.01X^2-0.1X+6

Determine the consumers' surplus if the wholesale market price is set at $4/disc. (Round your answer to two decimal places.)

$ ?

To determine the consumer's surplus, we need to calculate the area under the demand curve and above the market price line.

The demand function is given by the equation P = -0.01X^2 - 0.1X + 6, where P is the wholesale unit price in dollars and X is the quantity demanded each week, measured in units of a thousand.

We are given that the wholesale market price is set at $4 per disc. So, we need to find the quantity demanded at this price.

To do this, we set the demand function equal to the market price:

-0.01X^2 - 0.1X + 6 = 4

Rearranging the equation, we have:

-0.01X^2 - 0.1X + 2 = 0

Next, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:

X = (-b ± sqrt(b^2 - 4ac)) / (2a)

In the equation -0.01X^2 - 0.1X + 2 = 0, a = -0.01, b = -0.1, and c = 2. Plugging these values into the quadratic formula, we have:

X = (-(-0.1) ± sqrt((-0.1)^2 - 4(-0.01)(2))) / (2(-0.01))

Simplifying further, we get:

X = (0.1 ± sqrt(0.01 - 0.08)) / (-0.02)

X = (0.1 ± sqrt(0.01 + 0.08)) / (-0.02)

X = (0.1 ± sqrt(0.09)) / (-0.02)

X = (0.1 ± 0.3) / (-0.02)

Now we have two possible solutions:

X1 = (0.1 + 0.3) / (-0.02) = 0.2 / (-0.02) = -10

X2 = (0.1 - 0.3) / (-0.02) = -0.2 / (-0.02) = 10

Since X represents the quantity demanded, it cannot be a negative value. Therefore, the quantity demanded at a price of $4 per disc is 10 thousand units (X2).

Now we can calculate the consumer's surplus. The consumer's surplus is 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 is given by the demand curve.

To find the consumer's surplus, we need to find the area under the demand curve and above the market price line. Since the demand curve is a parabola, we can calculate the area using integration.

The formula for calculating the consumer's surplus is:

Consumer's Surplus = ∫[a,b] (Demand Function - Market Price) dX

In this case, the demand function is -0.01X^2 - 0.1X + 6 and the market price is $4, so we have:

Consumer's Surplus = ∫[0,10] (-0.01X^2 - 0.1X + 6 - 4) dX

Simplifying the integrand, we get:

Consumer's Surplus = ∫[0,10] (-0.01X^2 - 0.1X + 2) dX

Now we can integrate term by term:

Consumer's Surplus = [-0.01 * (X^3)/3 - 0.1 * (X^2)/2 + 2X] evaluated from 0 to 10

Plugging in the limits of integration, we have:

Consumer's Surplus = [-0.01 * (10^3)/3 - 0.1 * (10^2)/2 + 2 * 10] - [-0.01 * (0^3)/3 - 0.1 * (0^2)/2 + 2 * 0]

Simplifying further, we get:

Consumer's Surplus = [-1000/3 - 50 + 20] - [0/3 - 0/2 + 0]

Consumer's Surplus = -1000/3 - 50 + 20

Finally, calculating the numerical value, we have:

Consumer's Surplus ≈ -333.33 - 50 + 20 ≈ -363.33

Therefore, the consumer's surplus is approximately -$363.33.