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 = −0.01x2 − 0.3x + 10
Determine the consumers' surplus if the market price is set at $6/cartridge. (Round your answer to two decimal places.)
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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 = −0.01x2 − 0.3x + 56
Determine the consumers' surplus if the market price is set at $2/cartridge. (Round your answer to two decimal places.)
To determine the consumer's surplus, we need to calculate the area under the demand curve above the market price.
The formula for consumer's surplus is:
Consumer's Surplus = ∫[lower limit, upper limit] (Demand function - Price) dx
In this case, the lower limit is 0 (since we can't have negative quantity demanded) and the upper limit can be calculated by solving the demand function for x when the unit price (p) is $6.
Setting p = 6 in the demand function:
6 = -0.01x² - 0.3x + 10
Simplifying the equation:
0.01x² + 0.3x - 4 = 0
We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Plugging in the values for a, b, and c:
x = (-(0.3) ± √((0.3)² - 4(0.01)(-4))) / 2(0.01)
Simplifying:
x = (-0.3 ± √(0.09 + 0.16)) / 0.02
x = (-0.3 ± √0.25) / 0.02
x = (-0.3 ± 0.5) / 0.02
We consider only the positive solution since we can't have negative quantities demanded:
x = (-0.3 + 0.5) / 0.02 = 0.2 / 0.02 = 10
So the upper limit is 10 thousand units.
Now, let's calculate the consumer's surplus.
Consumer's Surplus = ∫[0, 10] (-0.01x² - 0.3x + 10 - 6) dx
Simplifying:
Consumer's Surplus = ∫[0, 10] (-0.01x² - 0.3x + 4) dx
Integrating term by term:
Consumer's Surplus = [-0.01 * (x³/3) - 0.3 * (x²/2) + 4x] from 0 to 10
Evaluating at the upper and lower limits:
Consumer's Surplus = [-0.01 * (10³/3) - 0.3 * (10²/2) + 4 * 10] - [-0.01 * (0³/3) - 0.3 * (0²/2) + 4 * 0]
Simplifying:
Consumer's Surplus = [-33.33 - 15 + 40] - [0 - 0 + 0]
Consumer's Surplus = -8.33
Therefore, the consumer's surplus is -$8.33.