Strongest Beam from a Log
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A wooden beam is to be cut from an 18-inch-diameter log as shown. It has a width, w, and a height, h.
The strength, S, of such a beam is
Equation 1
where k is a positive constant.
We wish to find the dimensions w and h which result in the strongest beam.
How should we proceed?
We can try to express S in terms of one variable and the constant k. Then we can determine the value of that variable which maximizes S in our usual way by differentiating, setting that result to zero, and solving for the value which maximizes.
How can we express S in terms of one variable?
We can use the geometry of the diagram to relate w and h.
Apply the Pythagorean Theorem. Note that the diagonal of the rectangular beam is equal to the 18-inch diameter.
Equation 2
Since
also appears in equation 1, solve for it in equation 2.
We get
Substitute in equation 1.
We get
Distribute the multiplication by w.
Equation 3
Set up the derivative.
Do the differentiation.
Equation 4
Solve for the value of w which maximizes the strength of the beam.
Take the square root of both sides.
Equation 5
Are we done?
No. We need to find the corresponding value of h.
How can we do that?
We can substitute equation 5 into equation 2. Do that.
We get
How can we check that this is a maximum and not a minimum?
We can find the second derivative of w from equation 3,
and evaluate it at w ≅ 10.4.
Then we can check the algebraic sign.
Do that.
What is the algebraic sign of the second derivative?
Since we have a positive value of w, the sign of the derivative is negative.
Is that consistent with a maximum?
Yes.
Draw a graph of equation 3 to also confirm that this is a minimum. Then check with the graph below.
Graph of strength versus width
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor