# math

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The stiffness S of a rectangular beam is proportional to its width (w) times the cube of its depth/height (h). find the dimensions (i.e.: w and h) of stiffest beam that can be cut from a log which has a circular cross-section with diameter of 12-in.

• math -

You would use the relation of the circular cross-section of (w/2)²+(h/2)²=(12/2)² to relate h and w.
After multiplying by 4 on both sides and rearranging, we have w=√(12²-h²).

Now the stiffness has been defined as
S=k(w)(h³) where k is a constant.

If you substitute w from above then S is now a function of h only.

Can you take it from here?

• math -

h^3=s/kw
h^3=s/k(12-h)

sorry, I'm still confused how to get the function of h. please help me to complete the assignment. thank you

• math -

You need to use the relation
w=√12²-h²) to eliminate w from the function S.

S(h)=k w h³
=k √12²-h²) h³

To find the maximum/minimum stiffness, you will proceed normally to equate
S'(h) = 0
to solve for h.

• math -

yes i understand now, thanks for your help

• math :) -

You're welcome!

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