I am a tutor and a year 8 student of mine has been given a maths test question that I can not fathom how to solve and explain for the year level he is at. It asks about a piece of luggage that has the requirements of the 3 dimensions totalling no more than 270cm when added together. It then asks what is the maximum possible volume a suitcase could have without exceeding this requirement. Optimisation generally involves calculus which would be far beyond this grade. The teachers answer is Vmax length=width=depth which means 90x90x90=729,000cm^3 she offers no information though as to how these 3 numbers were selected to give the optimum volume. What am I missing?? Or am I overthinking it?
[Math] Optimisation without calculus
geometryoptimizationvolume
Best Answer
This is simply a basic application of the AM-GM inequality.
If we label our dimensions $w,l,h$, we seek to maximize $wlh$ under the condition that $w+l+h=270$. Note that by AM-GM$$\frac{w+l+h}3\geq\sqrt[3]{wlh}$$$$wlh\leq90^3$$
But this implies that $w,l,h=90$ maximizes $wlh$.