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 surplus, we need to find the area of the triangle formed between the demand curve and the price line. The formula to calculate consumer surplus is:
Consumer Surplus = (1/2) * Base * Height
Step 1: Find the quantity demanded at a price of $4/disc.
Set the demand equation equal to the price:
-0.01x^2 - 0.1x + 6 = 4
Simplify the equation:
-0.01x^2 - 0.1x + 2 = 0
Step 2: Solve the quadratic equation for x values.
Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.01, b = -0.1, and c = 2.
x = (-(-0.1) ± √((-0.1)^2 - 4*(-0.01)*2)) / (2*(-0.01))
Simplify the equation:
x = (0.1 ± √(0.01 + 0.08)) / (-0.02)
Now, calculate the two possible values for x:
x₁ = (0.1 + √(0.09)) / (-0.02) = (0.1 + 0.3) / (-0.02) = -0.2 / (-0.02) = 10
x₂ = (0.1 - √(0.09)) / (-0.02) = (0.1 - 0.3) / (-0.02) = -0.2 / (-0.02) = 10
Since the quadratic equation has the same solution twice, there is only one value for x, which is 10.
Step 3: Calculate the consumer surplus.
Now, we can calculate the height of the triangle. Since the price is given as $4/disc, and the demand equation gives the price as a function of quantity, we can substitute the value of x (10) into the demand equation to find the corresponding price:
p = -0.01x^2 - 0.1x + 6
p = -0.01(10)^2 - 0.1(10) + 6
p = -0.01(100) - 1 + 6
p = -1 - 1 + 6
p = 4
The height of the triangle (or the vertical distance between the demand curve and the price line) is 4 - 0 (since the price line is at $4/disc and the demand curve intersects at 0).
The base of the triangle (or the quantity demanded) is x, which we found to be 10.
Now, we can calculate the consumer surplus using the formula:
Consumer Surplus = (1/2) * Base * Height
Consumer Surplus = (1/2) * 10 * 4
Consumer Surplus = 20
Therefore, the consumer surplus is $20.