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 − 0.1x + 32

Determine the consumers' surplus if the market price is set at $2/cartridge. (Round your answer to two decimal places.)

Demand function:  p = -0.01x^2-0.2x+13  (1).

The equation (2) is: CS = ∫ D(p) dp, where D is the demand curve expressed in terms of the price.  The bottom limit of the integrand is the market price given as $5 and the top limit is $13 found by setting x = 0 in (1) above. Equation (2) above requires we solve the original equation, (1) above, for x in terms of p.  The first step is (a) to multiply both sides of equation by -100 to clear decimals, then (b) place equation in standard quadratic form, namely, x^2 + 20x + 100p – 1300 = 0.  Step (c): Solve for x by applying quadratic formula, namely, x = (-b +- √(b^2 – 4ac)) / 2a to solve for x.  Use a = 1, b = 20, c = 100p – 1300. The new demand equation is: x = -10 +  10√(14 – p). Now calculate ∫ (-10 +  10(14 – p)^(1/2)) dp and evaluate at 5 and 13.   This gives: -10(13 – 5) – (2/3) (10) ((14 - 13)^(3/2) – (14 – 5)^(3/2)) .  This evaluates to: -80 – (20/3) + 180.   So, CS = $93.33.

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