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The demand function for a particular vacation package is $ p(x) = 2000 - 46 \sqrt{x} $ Find the consumer surplus when the sales level for the packages is 400. Illustrate by drawing the demand curve and identifying the consumer surplus as an area.
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Calculus 2 / BC
Chapter 8
Further Applications of Integration
Section 4
Applications to Economics and Biology
Applications of Integration
Campbell University
Harvey Mudd College
Baylor University
University of Michigan - Ann Arbor
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Okay, So the problem gives us following demand curve 2000 minus 46 square root of X and a quantity of 400 packages and asked us to find consumer surplus. So first I've sketched the graph and just so that we're all aware consumer surplus is going to be this area and green. So the first thing we need to d'oh is we need to find where on the p access the value of 400 corresponds to. So we take 2000 and subtract 46 times the square root of 400 which is 2000 minus 46 times 20 which is equal to 1000 80. So 400 corresponds to a price of $1080. No, to get the green area. What we're going to do is we're going to first take this blue region and then subtract off this red region. So for the blue, it's given by the integral from zero 400 of her function, which is 2000 minus 46 square e of x, the ex, and then to get her green consumer surplus, we're going to subtract away the red, which is just a rectangle with side lengths of 400 and 1080. So this to section it off is going to be 2000 2000 x from zero 4000 minus and now for the square root. Since we're adding one to the power, we get three halves, but we're dividing, so it's going to be two times 46 over three necks to the three halves from zero. 4000 and subtracting 432,000. So the good thing is that the zero terms cancel since there is zero input into zeroes and for ex would make them zero. So we only need to evaluate at 4000. So 2000 times 4000 going down here is going to be 800,000 and then the next term is 92 Over three times 1000 finally begins attracting off 432 1000. So because of this middle term, we're going to end up with an answer in 1/3 which, to be precise, is 126,666 and 2/3. And since we're talking about not 126 122,000. And since we're talking about dollar quantity, I'm going to round to their scent $220,666 and 67 sets. And that in green is the consumer surplus, which was the green on the graph.
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