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anonymous
 5 years ago
A manufacturer has been selling flashlights at $6 a piece, and at this price consumers have been buying 3000 flashlights per month. The manufacturer wishes to raise the price and estimates that for each $1 increase in price, 1000 fewer flashlights will be sold each month. The manufacturer can produce the flashlight a cost of $4 per flashlight. At what price should the manufacturer sell the flashlights to generate the greatest profit?
anonymous
 5 years ago
A manufacturer has been selling flashlights at $6 a piece, and at this price consumers have been buying 3000 flashlights per month. The manufacturer wishes to raise the price and estimates that for each $1 increase in price, 1000 fewer flashlights will be sold each month. The manufacturer can produce the flashlight a cost of $4 per flashlight. At what price should the manufacturer sell the flashlights to generate the greatest profit?

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Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1i need to go soon so I will leave a couple of suggestions, ok ?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0For one I don't know where to start

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1the rate that the people buy this item is 3000  1000(x6) since when the price is $6 , 3000 items are sold and for each $ increase from $6 the number of items sold decrease by 1000

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1this is where x is the price of the item

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1now Profit = Sales  Cost so we need to make a function of profit and maximize it. if we let Profit = p(x)

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1p(x) = (number of items sold) * (price of each item)  (total cost for manufacturing the items) so p(x) = {30001000(x6)} *x  (30001000(x6)) i

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1you can simplify this to p(x) = 1000* {3(x6)}*(x1) = 1000* (9x)*(x1)

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1to maximize this, you find p'(x) and find the local maximum. just in case check the boundary to see if the price was not the maximum.

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1sorry, (x1) is actually (x4) I forgot to multiply 4 for the cost. anyway, was that of any help ?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0It does indeed help but I don't understand how you figure 30001000(x6) is the rate of flashlights purchased

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1so when x=6, to make y = 3000 y has to be 30001000(x6)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I don't know what it is about it, I just don't understand how you formulated that function.

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1ok, when x =6, y = number of items sold = 3000 right ?

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1so, using \[yy_1 = m(xx_1)\]

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1y  3000 = 1000(x6) so y = 30001000(x6)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So do I take 1000* (9x)*(x4) and expand it then differentiate ?

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1if you don't mind the product rule of differentiation then you can just leave it like that. it does seem like it would be easier though. so yes, you let p'(x) = 0 and find the critical points. also check the endpoints because it is possible that when the price is cheapest or highest you get the maximum profit. (although in real life it doesn't happen)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.02x + 4 = f'(x) right?

Yuki
 5 years ago
Best ResponseYou've already chosen the best response.1sorry I gotta go :/ but the differentiation part should be easy enough. I'm pretty sure you can get it :) I am guessing that the price has to be slightly higher than 6.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Ok, thanks for the help!!
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