The element of a cone has length L. For what height H (with respect to L) will the volume of the cone be the largest?
This is a good exercise to introduce student to the idea of derivatives and their possible practical use.
Firstly, I should draw the cone and notice that its volume is equal to 1/3piH*R2 (where R - radius of the base of the cone). The next step is to eliminate one of the two unknowns (H or R). I need to calculate H, so R can be eliminated. To do it, I will use Pythagoras theorem and count R with respect to H and L. I need to remember that L is given so it doesn’t need to (and actually can’t) be calculated.
After applying the Pythagoras theorem and a simple substitution, I end up with a formula for the volume that is dependent on L and H, just as I expected. In order to see when the volume is largest, I will need to think of the formula as a function. After differentiating the formula with respect to H and equating it to zero, I can find stationary points of the function, i.e. values of H for maximum and minimum volumes of the cone. Knowing the principles of derivatives, it can be found that the volume of the cone is largest when H = L/sqrt(3)
