The demand function for a certain brand of CD is given by
p = −0.01x^2 − 0.1x + 14
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 $8/disc. (Round your answer to two decimal places.)
= −0.01x^2 − 0.1x + 14
x^2 + 10x -1400+ 800 = 0
x = -30
x = 20
-0.01(20)^2 -.1(20) + 14
= 8
−0.01x^2 − 0.1x + 14 on [0,20] -8(20)
-0.01x^3/3 -0.1x^2/2 + 14x on [0,20] -160
-80/3 -20 + 280-160
= 73.33
To determine the consumers' surplus, we need to find the area under the demand curve and above the market price line.
The market price is set at $8/disc, so we substitute this price into the demand function:
p = −0.01x^2 − 0.1x + 14
8 = −0.01x^2 − 0.1x + 14
Now we solve this equation to find the quantity demanded (x):
−0.01x^2 − 0.1x + 14 - 8 = 0
−0.01x^2 - 0.1x + 6 = 0
To solve this quadratic equation, we can either use the quadratic formula or factorization. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -0.01, b = -0.1, and c = 6. Substituting these values into the quadratic formula, we get:
x = (0.1 ± √((-0.1)^2 - 4*(-0.01)*6)) / (2*(-0.01))
Simplifying further, we have:
x = (0.1 ± √(0.01 + 0.24)) / (-0.02)
x = (0.1 ± √(0.25)) / (-0.02)
We have two possible solutions:
1) x = (0.1 + 0.5) / (-0.02) = -0.6 / (-0.02) = 30
2) x = (0.1 - 0.5) / (-0.02) = -0.4 / (-0.02) = 20
Since we are dealing with a quantity measured in units of a thousand, the quantity demanded is 30,000 units for a unit price of $8/disc.
To determine the consumers' surplus, we need to find the area under the demand curve and above the market price line. The consumers' surplus is given by the integral of the demand function from 0 to the quantity demanded (30,000 units):
Consumer Surplus = ∫[0, 30,000] (-0.01x^2 - 0.1x + 14) dx
Integrating this function, we get:
Consumer Surplus = [-0.01/3 * x^3 - 0.1/2 * x^2 + 14x] [0, 30,000]
Substituting the upper and lower limits, we have:
Consumer Surplus = [-0.01/3 * (30,000)^3 - 0.1/2 * (30,000)^2 + 14 * (30,000)] - [0]
Calculating this expression will give us the consumers' surplus.