Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies.
The question is asking us to find the "strongest" beam, thus we are optimizing strength, which is given by the formula . There are two variables here: width (w) and height (h). In order to simply the formula to one variable, we want to relate w and h using another formula. Looking at the diagram of the log, it would make sense to use the Pythagorean theorem, as w and h are the two side of a right triangle. Thus we have

Now we can solve this equation for w or h and substitute it back into our formula . Which variable should I choose? Either one will work, but because I have an in my strength formula and an in the Pythagorean theorem, I will solve for . Solving for in the Pythagorean theorem gives
.
Substituting this back into my strength formula gives

Which is now a function in one variable (w) since k is a constant. Note that because the width of the beam is constrained by the diameter of the log, the domain for this function is .
I know that my function S will be maximized at either the endpoints of the domain or at a critical point. The endpoints are and . To find critical points, I first find the derivative:

And then set S' equal to zero and solve.

The negative square root is not in our domain, so we have one critical point, .
I can check for the maximum in two ways. The first (and more rigorous) method is to plug the endpoints and critical point back into my function and compare the values I get.
So is maximized when . Recalling that we can solve for .
The second way to check for the maximum to realize that the endpoints will cause to be zero (same process as checking the endpoints above), and to perform the first (or second) derivative test to check that is a maximum.
|