mathcalculus
An orchard contains 51 peach trees with each tree yielding an average of 51 peaches. For each 4 additional trees planted, the average yield per tree decreases by 3 peaches. How many trees should be planted to maximize the total yield of the orchard?
The number of trees is . (Give your answer as a whole number.)
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mathcalculus
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I got my answer 17/.. but they want it as a whole number. any ideas what i might have done wrong?
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@agent0smith
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@pgpilot326
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@zepdrix
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17/2*
pgpilot326
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total peaches = (51 - (3/4) x)*(51+x) where x = # number of additional trees.
maximize and if it's not an integer, which ever integer is closest will do the job.
mathcalculus
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yes i did it exactly like that
pgpilot326
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and the max occured at 17/2?
mathcalculus
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then i multiplied them out and to maximize I found the derivative.
mathcalculus
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well i got f(x)= -3x^2/4+51x/4+2601
mathcalculus
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and f '(x)= -6x/4+51/4
mathcalculus
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i set that to zero and got 17/2
pgpilot326
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so check the yield at x = 8 and x = 9. they're likely the same so either would work. but if you're a farmer, you go with less additional trees to save $
pgpilot326
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by yield, i mean total yield, not yield per tree.
pgpilot326
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x = 8 => 2655 peaches
x = 9 => 2655 peaches
mathcalculus
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i tried those also
mathcalculus
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it says wrong
mathcalculus
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pgpilot326
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did you try 8?
mathcalculus
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yeah
mathcalculus
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:/
pgpilot326
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i don't know what to tell you. could it be they want 59 as the asnwer (total number of trees in the orchard)?
mathcalculus
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woahhhh how is it 59?
mathcalculus
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its saying correct to that one
mathcalculus
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did you use the second derivative test? or something?
pgpilot326
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51 + 8 = 59. there were 51 already planted plus the 8 you added to maximize the total yield.
mathcalculus
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but if it's 8.5 shouldnt we round it to 9?
mathcalculus
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so wouldnt it be 60?
pgpilot326
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question wasn't clear... should have said total number of trees planted in the orchard.
you have to think like a farmer. one less tree gives the same yield as one more. what would be better? think of not only the initial cost of the tree but also the watering and care, etc.
mathcalculus
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ok...
pgpilot326
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maybe 60 would also work (as an answer) but it's not as practical as 59.
mathcalculus
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im a little confused on that.
mathcalculus
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yeah i tried 60 but it kept saying wrong.
mathcalculus
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i mean mathematically speaking best round is 60 and then adding it to the already planted trees
mathcalculus
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meaning 51+ 9....
pgpilot326
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again, you have think like a business person/farmer.
think of the problem like this: if you had to cut off fingers to get gold coins in the same scenario, you'd want to cut off as few fingers as possible in order to maximize the number of gold coins.
if you got the same number of gold coins for cutting of 8 fingers as you would for cutting off 9 fingers, how many fingers would you cut off?
mathcalculus
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huh?
pgpilot326
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how old are you?
mathcalculus
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this example sounds scary.
pgpilot326
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yeah, intentionally.
mathcalculus
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i understand you, but i just don't like the example.lol
mathcalculus
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thanks anyways, i see the problem clearly now.
pgpilot326
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lol, sorry. it's just what popped out of my head.
mathcalculus
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thanks again!
pgpilot326
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excellent. ask your teacher to be more explicit in what is wanted next time.
you're welcome!