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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.

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

Answers by Expert:

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.

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

(See above.)