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.)

To determine the consumers' surplus, we need to integrate the demand function from zero to the quantity demanded at the given market price.

Given demand function: p = -0.01x^2 - 0.3x + 20

Market price: $2/cartridge

To find the quantity demanded at the market price, we can set p = 2 and solve for x:

2 = -0.01x^2 - 0.3x + 20

Simplifying this equation, we get:

0.01x^2 + 0.3x - 18 = 0

We can solve this quadratic equation by using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = 0.01, b = 0.3, and c = -18. Plugging in these values, we can solve for x:

x = (-0.3 ± √(0.3^2 - 4 * 0.01 * -18)) / (2 * 0.01)

x = (-0.3 ± √(0.09 + 0.72)) / 0.02

x = (-0.3 ± √0.81) / 0.02

x = (-0.3 ± 0.9) / 0.02

x = (-0.3 + 0.9) / 0.02 or x = (-0.3 - 0.9) / 0.02

x = 0.6 / 0.02 or x = -1.2 / 0.02

x = 30 or x = -60

Since quantity cannot be negative, we take x = 30 as the quantity demanded at the market price of $2/cartridge.

Now, we can calculate the consumers' surplus by integrating the demand function from zero to 30:

Consumers' Surplus = ∫[0, 30] (-0.01x^2 - 0.3x + 20) dx

To evaluate this integral, we need to find the antiderivative (or integral) of the demand function:

∫(-0.01x^2 - 0.3x + 20) dx = -0.01 * (x^3 / 3) - 0.3 * (x^2 / 2) + 20 * x + C

Now, we can substitute the limits of integration (0 and 30) into this antiderivative and calculate the consumers' surplus:

Consumers' Surplus = [-0.01 * (30^3 / 3) - 0.3 * (30^2 / 2) + 20 * 30] - [-0.01 * (0^3 / 3) - 0.3 * (0^2 / 2) + 20 * 0]

Consumers' Surplus = [-0.01 * (27000 / 3) - 0.3 * (900 / 2) + 600] - [0]

Consumers' Surplus = [(-270 / 3) - (270 / 2) + 600] - [0]

Consumers' Surplus = [-90 - 135 + 600] - [0]

Consumers' Surplus = 375

Therefore, the consumers' surplus when the market price is set at $2/cartridge is $375 (rounded to two decimal places).