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The sum of consumer surplus and producer surplus is called the $ total surplus $; it is one measure economists use as an indicator of the economic health of a society. Total surplus is maximized when the market for a good is in equilibrium.

(a) The demand function for an electronics company's car stereos is $ p(x) = 228.4 - 18x $ and the supply function is $ p_s (x) = 27x + 57.4 $, where $ x $ is measured in thousands. At what quantity is the market for the stereos in equilibrium?

(b) Compute the maximum total surplus for the stereos.

a) 3800 and the price is $\$ 160$

b) $\$ 324,900$

Applications of Integration

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Okay, so the problem here gives us a supply curve. Piece of S of X is equal to 27 x plus 57.4 and a demand function P of X equal to 22 point or 228.4 minus 18 x and part A assets. When does equilibrium occur at what number of units and then part be asked us to find total surplus? So report a the number of units were concerned with is going to be a value for X And so to get that at the delivery, um, we just set the two functions equal to each other. So 27 x plus 57.4 equal to 220 80 next now Agnes and subtracting 45. So that's it. Now, these thousands of years. So finally is at 3800 stereos and that is for part a. Now, for part B. It asked us to find total surplus. Now the great thing about total surplus is that you don't need to find equilibrium, ply price and split this area since total surplus is that entire area and since both the supply and demand functions are linear. We don't need two, um, do any integration. We could just use the formula for the area of a triangle. And when we do that, we get that our total surplus is equal to 1/2 times the base of our triangle, which is the difference between 28.4 and 57.4 times the height, which is our 3800. And we're using the 3800 here because these functions are in terms of thousands. And so rather than using the 3.8, which would give us an answer in terms of thousands of dollars if we just want an answer in terms of dollars, we use the full number of units. And so we have 1/2 times 1 71 times 3800 which, as our final result we get Total surplus is equal to 324,000 $900. And that is the answer for part B.