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 $1/cartridge. (Round your answer to two decimal places.)

To determine the consumers' surplus, we need to calculate the area under the demand curve up to the market price of $1/cartridge.

Step 1: Setting the market price
Given that the market price is $1/cartridge, we substitute p = 1 into the demand equation:
1 = -0.01x^2 - 0.3x + 19

Step 2: Solving the quadratic equation
Rearrange the equation to have a quadratic equation equal to zero:
0.01x^2 + 0.3x - 18 = 0

Next, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a

For our equation, a = 0.01, b = 0.3, and c = -18. Plugging these values into the quadratic formula, we get:
x = (-0.3 ± √(0.3^2 - 4 * 0.01 * -18)) / (2 * 0.01)

After simplification:
x = (-0.3 ± √(0.09 + 0.72)) / 0.02

Calculating further:
x = (-0.3 ± √0.81) / 0.02
x = (-0.3 ± 0.9) / 0.02

This gives us two possible values for x:
x₁ = (-0.3 + 0.9) / 0.02 = 30
x₂ = (-0.3 - 0.9) / 0.02 = -60

Since we are talking about quantity demanded, we discard the negative value. Thus, x = 30.

Step 3: Calculating the consumers' surplus
To find the consumers' surplus, we calculate the area under the demand curve (p(x)) from x = 0 to x = 30, subtracting the area of the triangle formed between the demand curve and the price line of $1.

Consumers' surplus = ∫[0, 30] (p(x) - 1) dx - (0.5 * (30 - 0) * (p(30) - 1))

Using the definite integral notation, the consumers' surplus can be calculated as:
Consumers' surplus = ∫[0, 30] (-0.01x^2 - 0.3x + 19 - 1) dx - (0.5 * (30 - 0) * ((-0.01(30)^2 - 0.3(30) + 19) - 1))

Simplifying, we have:
Consumers' surplus = ∫[0, 30] (-0.01x^2 - 0.3x + 18) dx - (0.5 * 30 * (-0.01(30)^2 - 0.3(30) + 18))

Now, we can calculate this definite integral using calculus or a software/tool that can perform symbolic integration, such as Wolfram Alpha. The definite integral will give us the consumers' surplus.

Once you have the result from the integral, round it to two decimal places to get the final value of the consumers' surplus.