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 consumers' surplus, we need to find the area between the price line and the demand curve up to the quantity demanded at the given wholesale market price.
Let's start by setting the wholesale market price, p, to $4/disc in the demand equation:
4 = -0.01x^2 - 0.1x + 6
Now we need to solve for x, which represents the quantity demanded at the given price. Rearranging the equation, we have:
0.01x^2 + 0.1x - 2 = 0
Next, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 0.01, b = 0.1, and c = -2. Plugging these values into the quadratic formula, we get:
x = (-0.1 ± √(0.1^2 - 4 * 0.01 * -2)) / (2 * 0.01)
Simplifying further:
x = (-0.1 ± √(0.01 + 0.08)) / 0.02
x = (-0.1 ± √0.09) / 0.02
x = (-0.1 ± 0.3) / 0.02
Now we have two possible values for x:
x1 = (-0.1 + 0.3) / 0.02 = 10
x2 = (-0.1 - 0.3) / 0.02 = -20
Since we're only interested in positive quantities, we can disregard the negative value. Therefore, the quantity demanded at a wholesale market price of $4/disc is 10 thousand units.
Next, let's find the consumers' surplus by calculating the area between the price line and the demand curve up to the quantity of 10 thousand units.
The consumers' surplus can be found by integrating the demand equation from 0 to 10, and then subtracting the result from the area of a rectangle with height 4 (the price) and width 10.
To integrate the demand equation, we need to find the antiderivative:
∫ (-0.01x^2 - 0.1x + 6) dx
= -0.01 * (x^3/3) - 0.1 * (x^2/2) + 6x + C
Now we can plug in the upper and lower limits of integration (0 to 10) into the antiderivative to find the definite integral:
[ -0.01 * (10^3/3) - 0.1 * (10^2/2) + 6 * 10 ] - [ -0.01 * (0^3/3) - 0.1 * (0^2/2) + 6 * 0 ]
Simplifying further:
[ -100/3 - 50 + 60 ] - [ 0 ]
= -100/3 + 10
= -90/3
= -30
Now, to find the total consumers' surplus, we subtract the result from the area of the rectangle:
Consumers' surplus = (4 * 10) - (-30)
= 40 + 30
= 70
Therefore, the consumers' surplus is $70.