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 - 0.1 x + 47
Determine the consumers' surplus if the market price is set at $5/cartridge. (Round your answer to two decimal places.)
To determine the consumer surplus, we need to find the area under the demand curve and above the market price. The formula for consumer surplus can be calculated by subtracting the total expenditure from the area under the demand curve.
The first step is to calculate the quantity demanded at the market price of $5/cartridge by substituting p=5 into the equation and solving for x:
5 = -0.01x^2 - 0.1x + 47
Rearranging the equation, we have:
0.01x^2 + 0.1x - 42 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 0.01, b = 0.1, and c = -42. Plugging these values into the formula, we get:
x = (-0.1 ± √(0.1^2 - 4 * 0.01 * -42)) / (2 * 0.01)
Simplifying the equation further, we have:
x = (-0.1 ± √(0.01 + 1.68)) / 0.02
x = (-0.1 ± √1.69) / 0.02
x = (-0.1 ± 1.3) / 0.02
Now, calculate the two possible values of x:
x1 = (-0.1 + 1.3) / 0.02 = 65
x2 = (-0.1 - 1.3) / 0.02 = -65
Since the quantity cannot be negative in this context, we discard the negative value of x, and the quantity demanded at the market price of $5/cartridge is 65.
Next, we calculate the area under the demand curve from x = 0 to x = 65. The formula for the area under the curve can be found by integrating the demand function with respect to x over this interval:
∫[0,65] (-0.01x^2 - 0.1x + 47) dx
The integral of -0.01x^2 with respect to x is -0.01 * (x^3/3).
The integral of -0.1x with respect to x is -0.1 * (x^2/2).
The integral of 47 with respect to x is 47x.
Integrating each term, we have:
-0.01 * (65^3/3) - 0.1 * (65^2/2) + 47 * 65
= -670 + 2117.5 + 3055
= 4502.5
So, the area under the demand curve from x = 0 to x = 65 is 4502.5.
Finally, we calculate the consumer surplus by subtracting the total expenditure (the market price multiplied by the quantity demanded) from the area under the curve:
Consumer Surplus = Area under the curve - Total Expenditure
Total Expenditure = Market price * Quantity demanded = $5 * 65 = $325
Consumer Surplus = 4502.5 - 325 = $4177.5
Therefore, the consumer surplus would be $4177.5 when the market price is set at $5/cartridge.