Answer:
Let's denote the diameter of the round log as \(D\) (which is twice the radius, \(r\)). The rectangular cross section of the beam is inscribed in the circle, meaning the height and width of the rectangle must fit within the diameter.
Given the problem, the strength \(S\) is proportional to the width \(w\) and the square of the height \(h\), expressed as \(S = kh^2w\).
For the rectangular cross section inscribed in a circle, the width \(w\) is equal to the diameter of the circle, and the height \(h\) is equal to the radius of the circle. Therefore, \(w = D\) and \(h = r\).
Now, substitute these values into the strength equation:
\[ S = k \cdot r^2 \cdot D \]
The diameter \(D\) is given as 44 centimeters, so \(D = 44\) cm. The radius \(r\) is half of the diameter, so \(r = \frac{D}{2} = 22\) cm.
Substitute these values into the equation:
\[ S = k \cdot (22)^2 \cdot 44 \]
To find the dimensions of the strongest beam, we want to maximize \(S\). Since \(S\) is directly proportional to \(r^2\) and \(D\), we maximize \(S\) by maximizing \(r\).
So, the strongest beam is a square beam with a side length equal to the radius of the log, i.e., \(h = r = 22\) cm and \(w = D = 44\) cm.
