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During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $ \$ 10 $ each and his sales averaged 20 per day. When he increased the price by $ \$ 1 $, he found that the average decreased by two sales per day.

(a) Find the demand function, assuming that it is linear.

(b) If the material for each necklace costs Terry $ \$ 6 $, what should the selling price be to maximize his profit?

a) $p(x)=20-\frac{1}{2} x$

b) $\$ 13$

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sells necklaces on the beach when he sells them for $10 he sells 20 a day. When he sells them for $11 he sells only 18 a day. So we want to find the demand function, which is also called the Price Function. I want to write that their price function. Assume that it's linear. And then, if it cost him $6 to make each one what price should be charged to maximize profits? Okay, so if it's a price function, we wanted to give us the price. So we want the price to be why and the numbers told to be X. All right, so first we'll find the slope. So Slope is why minus y over X minus x so minus one half. And then we'll use that to write the equation. So slope equals Y minus y over X minus X cross Multiply negative X plus 20 equals two Y minus 20. So negative website negative X plus 40 equals two y or negative one half X plus 20 equals y. Okay, so that's the price that he should charge. Or that's the price he's charging. If he sells ex things, then that will tell us what? The prices. All right, So now we want to know how he should maximize his profit. So we need to write a profit equation. And profit will equal the A revenue How much money he makes, minus the cost. How much he spent. So revenue is pricey charges price charged times number sold, minus price to make times number souls. Okay, so profit equals. He's going to charge this much minus one a half X plus 20. And he's going to sell X of them. And they cost $6 to make one. So six X to make X of them. So the prophet minus one half X squared plus 14 x. All right, so we're trying to maximize that. So we're gonna take its derivative. Yeah. P prime is minus X plus 14. Set that equal to zero. So if he sells 14 of them, he will make the maximum profit. How much? How much should he charge to do that? We'll remember Pete P of X little P of X is our equation to minus one half X plus 20 so P of 14 minus one half times 14 plus 20 minus seven plus 20th $13. So if he charges $13 he'll sell 14 of them and he will not make the maximum profit okay, If he charges less, he will sell more. But he won't make as much profit. If he charges more, he'll sell less, but he still won't make as much profit, so

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