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 $9/cartridge. (Round your answer to two decimal places.)
To determine the consumer surplus at a market price of $9/cartridge, we need to calculate the area under the demand curve up to this price.
The consumer surplus can be seen as the difference between the maximum amount consumers are willing to pay and the actual price they pay. It represents the benefit consumers receive from paying less than what they are willing to pay.
To find the consumer surplus, we first need to calculate the quantity demanded at the price of $9/cartridge.
Given that the demand function is p = -0.01x^2 - 0.3x + 19, we can substitute p = 9 into the equation and solve for x:
9 = -0.01x^2 - 0.3x + 19
Rearranging the equation and setting it to zero:
0.01x^2 + 0.3x - 10 = 0
To solve this quadratic equation, you can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, the coefficients are a = 0.01, b = 0.3, and c = -10. Plugging them into the formula:
x = (-0.3 ± sqrt(0.3^2 - 4 * 0.01 * -10)) / (2 * 0.01)
Calculating this equation will give you two solutions for x. One of them will be negative, which is not meaningful in this context. The positive solution will represent the quantity demanded at the price of $9/cartridge.
Once you find the value of x, you can substitute it back into the demand function to determine the actual quantity demanded.
With the quantity demanded and the market price, you can now calculate the consumer surplus.
Consumer surplus = 0.5 * (quantity demanded) * (market price - minimum price)
The minimum price is obtained by substituting the quantity demanded into the demand function and solving for p.
Finally, round the calculated consumer surplus to two decimal places.