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 consumers' surplus, we need to find the area between the demand curve and the market price line. In this case, the market price is set at $5/cartridge.
Step 1: Find the quantity demanded at the market price.
To find the quantity demanded, we substitute the market price, p = $5, into the demand equation and solve for x.
p = -0.01x^2 - 0.1x + 47
5 = -0.01x^2 - 0.1x + 47
0 = -0.01x^2 - 0.1x + 42
Step 2: Solve for x using the quadratic formula.
The equation -0.01x^2 - 0.1x + 42 = 0 is a quadratic equation. We can solve it using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = -0.01, b = -0.1, and c = 42.
x = (-(-0.1) ± √((-0.1)^2 - 4*(-0.01)*42))/(2*(-0.01))
x = (0.1 ± √(0.01 + 1.68))/(-0.02)
x = (0.1 ± √1.69)/(-0.02)
Taking the positive root:
x = (0.1 + √1.69)/(-0.02)
Step 3: Calculate consumers' surplus.
Consumers' surplus is the area between the demand curve and the market price line.
To calculate this area, we need to find the definite integral of the demand function from zero to the quantity demanded at the market price.
∫[0 to x](p dx) = ∫[0 to x](-0.01x^2 - 0.1x + 47) dx
Evaluating the integral:
∫[0 to x](-0.01x^2 - 0.1x + 47) dx = [-0.01(x^3/3) - 0.1(x^2/2) + 47x] [0 to x]
Plugging in the value of x:
[-0.01(x^3/3) - 0.1(x^2/2) + 47x] [0 to (0.1 + √1.69)/(-0.02)]
Evaluate the integral at the upper limit:
[-0.01((0.1 + √1.69)/(-0.02))^3/3 - 0.1((0.1 + √1.69)/(-0.02))^2/2 + 47((0.1 + √1.69)/(-0.02))]
Now, substitute the value of x = (0.1 + √1.69)/(-0.02) into the equation and calculate the consumers' surplus using a calculator or software.
After calculating, round the answer to two decimal places to get the consumers' surplus.