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Problem 10 Medium Difficulty

A camera company estimates that the demand function for its new digital camera is $ p(x) = 312e^{-0.14x} $ and the supply function is estimated to be $ p_s (x) = 26e^{0.2x} $, where $ x $ is measured in thousands. Compute the maximum total surplus


$\$ 996,792$


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Video Transcript

Okay, So in this problem, we're given a supply curve piece of SFX, which is 26 e to 0.2 x and demand curve P of X, which is 3 12 e to the negative 0.14 x and were asked to find the maximal total surplus so total surpluses maximized when we're at equilibrium. So first we need to find our equilibrium quantity. To do this, we just set our supply and demand curves equal to each other. When we do that, we have 26 e to 0.2 x equal to 312 e to the minus 0.14 x now dividing by E to the minus 0.14 x and dividing by 26 we get e to 0.34 X is equal to 12 now, taking the natural long on both sides, we get that 0.34 x equals the natural log of 12 and so X is the natural law of 12 over 0.34 Now for total surplus, we're going to first identify it on the graph in blue to find this blue region First, we're going to integrate the demand curve from zero to our equilibrium quantity, and that will give us the area shaded in green. Then we integrate this fly curve from zero to our equilibrium quantity, and that will give us the red area and taking the difference of these two will give us our total surplus in blue. So labeling it, we have our total surplus equal to the integral from zero two the natural log 12 over 0.34 and then 312 e to the minus 0.1 for X DX minus in the area and red, which is thean tha girl from zero to Ellen, 12 over 0.34 of 26 e 0.2 x d. X Now to evaluate the exponential inner rules. Since there's a coefficient in front of X, all we're going to do is divide by that coefficient, but then keep the exponential turn the same. So we will get minus 3 12 over 0.14 times E to the minus 0.14 x zero to Ellen, 12 over 0.34 and that minus same idea Over here, we'll take 27 divided by that coefficient on X for 0.2 on antique, he to the 0.2 X evaluate at zero and Ellen 12 over 0.34 So when you evaluate and plug in Ellen 12 over 0.34 in and then plug in zero for Ex, either zero will be one. And so, after evaluating both terms, we'll get 3 12 over 0.14 times, minus 0.641 and then subtract 26 over 0.2 times three point 313 which will give us 99 6.792 And then in the question it says that these graphs and these functions are for thousands. So, really, we should have multiplied through by 1000 which we can do right here in the end, when we say finally, our total surplus is approximately 996,792 dollars

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