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I've got this question from a textbook that doesn't show the proof. I would like to...

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konkonz | (Level 1) Honors

Posted May 24, 2013 at 10:11 AM via web

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I've got this question from a textbook that doesn't show the proof. I would like to know the proof to this question please.

A orchadist notices that an apple tree will produce 300 apples per year if 16 trees are planted in every standard-sized field. For every additional tree planted in the standard-sized field, she finds that the yield per tree decreases by 10 apples per year.

a) If she plants an additional 'x' trees in every standard-sized field, show that the total number of apples produced will be N= -10x^2 + 140x + 4800

b) How many trees should be planted in each field in order to maximize the number of apples that are produced?

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crmhaske | College Teacher | (Level 3) Associate Educator

Posted May 24, 2013 at 12:24 PM (Answer #1)

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The number of trees (N) that the orchard produces will be equal to the number of tress (T) multiplied by the yield per tree (Y):

`N=TY`

We know that the number of trees is equal to 16 plus how ever many more trees are planted:

`T=16+x`

We also know that the yield per tree is equal to 300 minus -10 multiplied by the number of trees that are added:

`Y=300-10x`

Therefore:

`N=(16+x)(300-10x)`
`=4800-160x+300x-10x^2`
`=-10x^2+140x+4800`

In order to determine the number of trees that should be planted in order to maximize the yield, we have to find for which value of x the derivative is equal to 0:

`N=-20x+140=0`
`20x=140`
`x=7`

Therefore, 7 additional trees should be planted in order to maximize the yield of the orchard.

Sources:

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konkonz | (Level 1) Honors

Posted May 24, 2013 at 1:22 PM (Reply #1)

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Thanks for your answer. You got a) correct and I want to thank you, but you got b) incorrect. The answer in the textbook says it is 23. Not 7.

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konkonz | (Level 1) Honors

Posted May 24, 2013 at 1:50 PM (Reply #2)

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Please disregard my previous message. You were right, I just forgot to factor in the 16 existing trees. Thanks!!!!

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crmhaske | College Teacher | (Level 3) Associate Educator

Posted May 24, 2013 at 2:55 PM (Reply #3)

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And I forget the word additional.  7 *additional* trees :).  Your welcome.

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konkonz | (Level 1) Honors

Posted May 24, 2013 at 11:18 PM (Answer #2)

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Just a quick question if you don't mind, I was taught to use x= -b/2a and then substitute that back in to the original equation to find minimum or maximum value of a quadratic function but that doesn't seem to work here in this case; as my x=7, but my y= some crazy number. Is there a reason to this?

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crmhaske | College Teacher | (Level 3) Associate Educator

Posted May 25, 2013 at 2:27 AM (Answer #3)

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You asked:

Just a quick question if you don't mind, I was taught to use x= -b/2a and then substitute that back in to the original equation to find the minimum or maximum value of a quadratic function but that doesn't seem to work here in this case; as my x=7, but my y= some crazy number. Is there a reason to this? - For the Apple Tree question. Thanks.

 

And I answered:

If you take the derivative of the standard form a second degree polynomial and set it to zero you will find that:

y=ax^2+bx+c
y'=2ax+b=0
2ax=-b
x=-b/2a

So as you can see, your method is the same as mine :)

For this question all you are asked to find is the value of x that maximizes y, or in this case, N. You don't need to substitute it back into the original equation because you aren't asked to indicate what the maximum yield of the orchard is at that number of additional trees. But, if you did so you would find that:

N=-10(7^2)+140(7)+4800=-490+980+4800=5290

So, if an additional 7 trees are planted, for a total of 16+7=23 trees, the orchard will yield a maximum of 5290 apples.

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And as promised, here is that graph of the function showing the maximum point at (7,5290):

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