Student Resources
Optimization: Strength of a beam
The strength of a beam is proportional to its width, \(w\), and the square of its height, \(h\). That is,
\(s=kwh^2\)
for some constant \(k\).
Below is a diagram of the cross-section of a beam that can be cut from a 30 cm diameter log. You can adjust the dimensions of the beam by moving the red point and use the results to answer the questions below.
A 20 cm high beam is cut from a 30 cm diameter log. Which of the following would have produced a stronger beam? Hint: Set the height to about 20 cm in the diagram above and see how you have to manipulate it to increase the strength.
Sketch a graph of the strength of a beam (cut from a 30 cm diameter log) as a function of its width, that is a graph of \(s(w)\).
Sketch a graph of the strength of a beam (cut from a 30 cm diameter log) as a function of its height, that is a graph of \(s(h)\).
Let \(w\) be the width of the rectangular cross-section of a beam cut from a 30 cm diameter log, and let \(h\) be the largest possible corresponding height that can be cut from the log, as illustrated in the interactive diagram. Find an equation relating \(h\) and \(w\) would allow you to solve for one of these two variables given the other one.
Find a formula for the strength of a beam (cut from a 30 cm diameter log) as a function of its width, that is a formula for \(s(w)\).
Find a formula for the strength of a beam (cut from a 30 cm diameter log) as a function of its height, that is a formula for \(s(h)\).
What is the approximate width (in cm) of the strongest beam that can be cut from a 30 cm diameter log?