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 + 19
Determine the consumers' surplus if the market price is set at $9/cartridge. (Round your answer to two decimal places.)
To determine the consumer surplus, we first need to find the quantity demanded when the unit price is $9.
The demand function is given by:
p = -0.01x^2 - 0.3x + 19
Substituting p = 9 into the equation, we can solve for x:
9 = -0.01x^2 - 0.3x + 19
Rearranging the equation:
0.01x^2 + 0.3x - 10 = 0
Now we can solve this quadratic equation using the quadratic formula, which states that for an equation ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 0.01x^2 + 0.3x - 10 = 0, a = 0.01, b = 0.3, and c = -10. Plugging these values into the quadratic formula, we get:
x = (-(0.3) ± √((0.3)^2 - 4(0.01)(-10))) / (2(0.01))
Simplifying further:
x = (-0.3 ± √(0.09 + 0.4)) / 0.02
x = (-0.3 ± √0.49) / 0.02
x = (-0.3 ± 0.7) / 0.02
There are two possible solutions:
x1 = (-0.3 + 0.7) / 0.02 = 0.4 / 0.02 = 20
x2 = (-0.3 - 0.7) / 0.02 = -1 / 0.02 = -50
Since the quantity demanded cannot be negative, we discard x2 = -50. Therefore, when the price is $9, the quantity demanded is 20 thousand units.
To calculate the consumer surplus, we need to find the area between the demand curve and the price line. In this case, since the demand curve is a downward-sloping parabola, we can use the formula for the area of a triangle to calculate the consumer surplus.
The formula for the area of a triangle is:
Area = (base * height) / 2
The base of the triangle is the quantity demanded, which is 20 thousand units. The height of the triangle is the difference between the price (given as $9) and the demand curve at that quantity.
Substituting the values into the formula:
Area = (20 * 10^3 * (19 - 9)) / 2
Area = (20 * 10^3 * 10) / 2
Area = 200,000
Therefore, the consumer surplus when the market price is set at $9/cartridge is $200,000.