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May 11th, 2011, 01:37 PM   #1
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Smallest area and Smallest Length of String around Box

I have a problem!

I have attached it to this post, just open the attachment please.
Attached Files
 CD1_SolsMan_Ch4.doc (64.0 KB, 30 views)

 May 11th, 2011, 10:38 PM #2 Newbie   Joined: May 2011 Posts: 5 Thanks: 0 Re: Smallest area and Smallest Length of String around Box Can anyone give me a few pointers????
 May 11th, 2011, 11:34 PM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Smallest area and Smallest Length of String around Box We have a rectangular box of a given volume V. The dimensions of the box can be labeled x, y, and z. The volume is then: $V=xyz$ The length L of string wrapped around the box (as in your second diagram) is: $L=2x+2z+2y+2z=2x+2y+4z$ Now, if x = y as in your second diagram, we have: (1) $V=x^2z$ (2) $L=4x+4z$ Solving (1) for z, we find $z=\frac{V}{x^2}$ and substituting for z into z, we get: $L=4x+4\frac{V}{x^2}$ Differentiating L with respect to x yields: $\frac{dL}{dx}=4-8Vx^{-3}=\frac{4$$x^3-2V$$}{x^3}=0$ $x^3=2V$ $x=$$2V$$^{\frac{1}{3}}$ Since $\frac{d^2L}{dx^2}=24x^{-4}>0=$ for x > 0, we know the critical value found gives a minimum. Thus, the dimensions of the box are: $x=y=$$2V$$^{\frac{1}{3}}$, $z=\frac{V}{$$\(2V$$^{\frac{1}{3}}\)^2}=$$\frac{V}{ 4}$$^{\frac{1}{3}}=\frac{x}{2}$
May 12th, 2011, 12:37 AM   #4
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Re: Smallest area and Smallest Length of String around Box

how can i apply this to the first diagram??

Quote:
 The first part of the question asks me to find the length of string in terms of width and Volume?

May 12th, 2011, 12:43 AM   #5
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Re: Smallest area and Smallest Length of String around Box

Sorry, maybe this may make it clearer

Open the following attachment, the diagram is hoppefully clearer...

Thankyou anyone who helps. I really appreciate it!!
Attached Files
 I have a problem.docx (25.0 KB, 24 views)

 May 12th, 2011, 02:37 AM #6 Newbie   Joined: May 2011 Posts: 5 Thanks: 0 Re: Smallest area and Smallest Length of String around Box can anyone help me please????

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