The demand function for a certain make of replacement cartridges for a water purifier is given by the following equation where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.
p=-0.01x^2-0.3x+20
Determine the consumers' surplus if the market price is set at $2/cartridge. (Round your answer to two decimal places.)
707
To determine the consumers' surplus, we need to calculate the area of the triangle bounded by the demand curve and the horizontal axis. The formula for consumers' surplus is given by:
Consumers' Surplus = ∫[0, x_0] (p(x) - p_m) dx
Where:
- p(x) is the unit price as a function of the quantity demanded (demand function)
- p_m is the market price
- x_0 is the quantity demanded at the market price
In this case, the market price is set at $2/cartridge. We need to find x_0, the quantity demanded at the market price.
Setting p(x) equal to p_m and solving for x, we have:
-0.01x^2 - 0.3x + 20 = 2
Simplifying the equation:
-0.01x^2 - 0.3x + 18 = 0
To solve this quadratic equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = -0.01, b = -0.3, and c = 18.
Applying the quadratic formula, we have two roots:
x = (-(-0.3) ± √((-0.3)^2 - 4(-0.01)(18))) / (2(-0.01))
Simplifying further:
x = (0.3 ± √(0.09 + 0.72)) / (-0.02)
x = (0.3 ± √0.81) / (-0.02)
x = (0.3 ± 0.9) / (-0.02)
Therefore, the solutions for x are:
1. (0.3 + 0.9) / (-0.02) = -60
2. (0.3 - 0.9) / (-0.02) = 30
Since the quantity demanded cannot be negative, we discard the negative solution. Hence, x_0 = 30.
Now we can calculate the consumers' surplus using the definite integral:
Consumers' Surplus = ∫[0, 30] (-0.01x^2 - 0.3x + 20 - 2) dx
Simplifying the equation within the integral:
Consumers' Surplus = ∫[0, 30] (-0.01x^2 - 0.3x + 18) dx
Integrating term by term:
Consumers' Surplus = [-0.01 * (1/3) * x^3 - 0.3 * (1/2) * x^2 + 18x] evaluated from 0 to 30
Evaluating the integral:
Consumers' Surplus = [-0.01 * (1/3) * 30^3 - 0.3 * (1/2) * 30^2 + 18 * 30] - [-0.01 * (1/3) * 0^3 - 0.3 * (1/2) * 0^2 + 18 * 0]
Consumers' Surplus = [-0.01 * 9000 - 0.3 * 450 + 540] - [0 - 0 + 0]
Consumers' Surplus = [-90 - 135 + 540] - [0]
Consumers' Surplus = 315
Therefore, the consumers' surplus is $315 (rounded to two decimal places) when the market price is set at $2/cartridge.