anonymous
  • anonymous
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?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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yuki
  • yuki
ready ?
yuki
  • yuki
i need to go soon so I will leave a couple of suggestions, ok ?
anonymous
  • anonymous
Ok

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anonymous
  • anonymous
For one I don't know where to start
yuki
  • yuki
the rate that the people buy this item is 3000 - 1000(x-6) 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
  • yuki
this is where x is the price of the item
yuki
  • yuki
now Profit = Sales - Cost so we need to make a function of profit and maximize it. if we let Profit = p(x)
yuki
  • yuki
p(x) = (number of items sold) * (price of each item) - (total cost for manufacturing the items) so p(x) = {3000-1000(x-6)} *x - (3000-1000(x-6)) i
yuki
  • yuki
you can simplify this to p(x) = 1000* {3-(x-6)}*(x-1) = 1000* (9-x)*(x-1)
yuki
  • yuki
to 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
  • yuki
sorry, (x-1) is actually (x-4) I forgot to multiply 4 for the cost. anyway, was that of any help ?
anonymous
  • anonymous
It does indeed help but I don't understand how you figure 3000-1000(x-6) is the rate of flashlights purchased
yuki
  • yuki
3000 is the y- intercept
yuki
  • yuki
-1000 is the slope
yuki
  • yuki
so when x=6, to make y = 3000 y has to be 3000-1000(x-6)
anonymous
  • anonymous
I don't know what it is about it, I just don't understand how you formulated that function.
yuki
  • yuki
ok, when x =6, y = number of items sold = 3000 right ?
anonymous
  • anonymous
Yes
yuki
  • yuki
when x = 7, y = 2000
anonymous
  • anonymous
Yes
yuki
  • yuki
so, using \[y-y_1 = m(x-x_1)\]
yuki
  • yuki
y - 3000 = -1000(x-6) so y = 3000-1000(x-6)
anonymous
  • anonymous
Ok, I see it, lol
anonymous
  • anonymous
So do I take 1000* (9-x)*(x-4) and expand it then differentiate ?
yuki
  • yuki
if 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
  • anonymous
-2x + 4 = f'(x) right?
anonymous
  • anonymous
woops, 13, not 4
yuki
  • yuki
sorry 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
  • anonymous
Ok, thanks for the help!!
yuki
  • yuki
good luck b

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