1. The demand function for a certain make of replacement cartridges for a water
ID: 3343489 • Letter: 1
Question
1.
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.
Determine the consumers' surplus if the market price is set at $8/cartridge. (Round your answer to two decimal places.)
2.
The demand function for a certain brand of CD is given by the following equation where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.
3.
It is known that the quantity demanded of a certain make of portable hair dryer is x hundred units/week and the corresponding wholesale unit price (in dollars) is the following.
Explanation / Answer
p(x) = .01x^2 - .2x + 54x = demand for a week, in units of 1000
Price is set at $6
Refer to diagram in Wikipedia article on Economic Surplus (link is below) to determine what the integral is supposed to be
The lower limit of the definite integral is zero (the leftmost, lowest x value for the area)
To find the upper limit of the definite integral (the upper x value), solve for x:
6 = -.01 x^2 - .2x + 54 Multiply through by 100 to make everything integers so we can see it better
600 = -x^2 - 20 x + 5400
-x^2 - 20 x + 4800 = 0
x^2 + 20 x - 4800 = 0
Complete the square (or your alternate favorite quadratic solving method. Personally I hate to switch windows because I lose the tab)
x^2 + 20 x + 100 - 100 - 4800 = 0
(x + 10)^2 - 4900 = 0
(x + 10)^2 = 4900
x + 10 = +/- 70
x = +70 - 10 = 60 <- use this one
x = -70 - 10 = -80 <- not feasible
Integrate for x = 0 to 60 p(x)-6 dx
Integral from 0 to 60 -0.1x^2 - .2x + 48
= -0.01/3 x^3 - 0.1 x^2 + 48x evaluated at 60, minus evaluated at 0 (which is zero, so just evaluate at 60)
-0.01/3 * 60^3 - 0.1 * 60^2 + 48*60
60 * (48 - 0.1 * 60 - 0.01 * 60^2 / 3) (There are definitely quicker ways to get this computed)
60 * (48 - 6 - 36/3)
60 * (48 - 6 - 12)
60 * 30 = 1800 answer http://en.wikipedia.org/wiki/Economic_su%u2026
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