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QUESTION: The national forest service finds that 16 trees are planted on an acre of land will grow approximately 3 feet per year. For each additional tree planted, the growth will be reduced by 0.5 inch. Find the number of tress per acre that will yield the largest amount of timber.

Kindly, tell me how to solve this question. thank you

ANSWER: Questioner:Osama
Country:Sindh, Pakistan
Category:Calculus
Private:No
Subject:Optimization Calculus

Question:
The national forest service finds that 16 trees are planted on an acre of land will grow approximately 3 feet per year. For each additional tree planted, the growth will be reduced by 0.5 inch. Find the number of tress per acre that will yield the largest amount of timber.

Kindly, tell me how to solve this question. thank you
............................................

Variables:

Growth rate of trees = G
Number planted per acre over 16 = x

Now find a way to express G in terms of x.

In general, the growth is Number of trees * growth-per-tree

So  G = (16 + x)(3 - 0.5x)

Take it from there.  You know the drill.  If not, see:

http://en.allexperts.com/q/Calculus-2063/2009/11/Maximum-minimum-problem-41.htm

for a collection of these problems I have answered for others


---------- FOLLOW-UP ----------

QUESTION: When I take derivative of it,
I got (-x-5) and when I put First derivative = 0,
I got -x-5=0
==> x= -5, what does it mean, when I got answer in negative. I had issue here.

Answer
Question: The national forest service finds that 16 trees are planted on an acre of land will grow approximately 3 feet per year. For each additional tree planted, the growth will be reduced by 0.5 inch. Find the number of tress per acre that will yield the largest amount of timber.

Kindly, tell me how to solve this question. thank you
............................................

Variables:

Growth rate of trees = G
Number planted per acre over 16 = x

Now find a way to express G in terms of x.

In general, the growth is Number of trees * growth-per-tree

So  G = (16 + x)(3 - 0.5x)

Take it from there.  You know the drill.  If not, see:

http://en.allexperts.com/q/Calculus-2063/2009/11/Maximum-minimum-problem-41.htm

for a collection of these problems I have answered for others


---------- FOLLOW-UP ----------

QUESTION: When I take derivative of it,
I got (-x-5) and when I put First derivative = 0,
I got -x-5=0
==> x= -5, what does it mean, when I got answer in negative. I had issue here.

==================================================
Yes, indeed, you would have an issue.  I misread one of the items.

It did say "grow approximately 3 FEET per year" and "will be reduced by 0.5 INCH".

In that case, I should have written:

So  G = (16 + x)(36 - 0.5x)   <<< 36 inches, not 3 inches.  Sorry.

Try it now.

[Actually, I think the problem-maker was rather careless in setting this problem.  Isn't the amount of lumber a CUBIC function of the height of the tree?  Don't get me started.]  

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Paul Klarreich

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All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

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I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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