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Calculus/Maximum-minimum problem

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Hi, I have the following problem I have to be able to answer in detail:
For a package, the length can be no more than 108 inches, and the length + girth (2(w+h)) can be no more than 130 inches. If the width of a package is 4 inches more than its height and it has the maximum length plus girth allowed, find the length that produces maximum volume.

So far I know the length = 108in, 2(w+h)=130in, and h=w+4. I just don't know where to go from here.

Answer
Questioner:Bibi
Country:Vermont, United States
Category:Calculus

Question:Hi, I have the following problem I have to be able to answer in detail:
For a package, the length can be no more than 108 inches, and the length + girth (2(w+h)) can be no more than 130 inches. If the width of a package is 4 inches more than its height and it has the maximum length plus girth allowed, find the length that produces maximum volume.

So far I know the length = 108in, 2(w+h)=130in, and h=w+4. I just don't know where to go from here. <<<<< Well, not exactly.  You don't know the length.

.............................
L + 2(w+h) = 130

And h = w + 4

Volume = L w h

Now you want to maximize Volume:

General scheme.  

There is something to be maximized (or minimized) -- the OPTIMIZABLE.
There is some constraint -- a condition that prevents you from making the OPTIMIZABLE (volume, here) gigantic.

1. Decide what the variables are -- the things that change.  Give them names.

2. Express the optimizable as a function of the variables.  At this point, you may have too many variables, so....

3. Use the constraint(s) as a way to eliminate all but one.

4. Run the machine.

OK, go to work:

Item 1 :Variables:
the volume is V
length is L
width is w
height is h

Item 2:  V = Lwh

Item 3:  But h = w + 4,

So V = Lw(w + 4)

and L + 2(w+h) = 130

Solve it:
L = 130 - 2(w + h)
L = 130 - 2(w + w + 4)
L = 130 - 2(2w + 4)
L = 130 - 4w - 8
L = 122 - 4w

Substitute back:
V = Lw(w + 4)

V = (122 - 4w)w(w + 4)

OKAY, then -- you have V  as a function of w. (a cubic in this case)
Item 4 - Run the machine.
Multiply out.
Differentiate.
Set it = 0.
Solve.
You know the drill -- take it from here.

Calculus

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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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