Two shapes both have a surface are of 1200cm2. One of them is a cylinder and one of them is a square based prism.
a) What is the maximum value of the volume of the shape if it is a cylinder?
b) What is the maximum value of the volume of the shape if it is a square based prism?
c) Which shape should you chose for a container you are building if you want the greatest possible volume and the least possible surface are?
The area and volume of a cylinder are respectively $$ A_{cylinder} = 2\pi r h + 2\pi r^2 \quad V_{cylinder} = \pi r^2 h $$ Where $r$ denotes the radius of the cylinder and $h$ denotes the height of the cylinder. Since we know that $A_{cylinder} = 1200$ we can express $h$ in terms or $r$ $$ h = \frac{A_{cylinder} - 2\pi r^2}{2\pi r} $$ Now let's use this to find an expression for the volume in terms of the surface area. $$ V_{cylinder} = \pi r^2 \frac{A_{cylinder} - 2\pi r^2}{2\pi r} = \frac{A_{cylinder}r-2\pi r^3}{2}$$ To find the maximum value we differentiate this expression and set it equal to zero. $$ \frac{dV_{cylinder}}{dr} = \frac{A_{cylinder}}{2}-6\pi r^2 = 0 $$ Thus we conclude that $r^2 = \frac{A_{cylinder}}{12\pi}$ and thus $r = \sqrt{\frac{A_{cylinder}}{12\pi}}$ since the radus cannot be negative. Putting it all together yields $$ V_{cylinder} = \frac{5A_{cylinder}}{6}\sqrt{\frac{A_{cylinder}}{12\pi}} = \frac{10000}{\sqrt{\pi}} cm^3$$