Question

A cylindrical-shaped cup has a height of 7 centimeters and a volume of 112 cubic centimeters. Henry fills the cup completely full of water. He then pours the water from the cup and completely fills a cone. If the cone has the same radius as the cup, what is the height of the cone?

Answer 1

let's bear in mind that the cylinder and the cone both have the same volume of 112 cm³, and the same radius, but different heights.

`\bf \textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\ \cline{1-1} V=112\\ h=7 \end{cases}\implies 112=\pi r^2(7)\implies \cfrac{112}{7\pi }=r^2\implies \cfrac{16}{\pi }=r^2 \\\\\\ \sqrt{\cfrac{16}{\pi }}=r\implies \cfrac{√(16)}{√(\pi )}=r\implies \cfrac{4}{√(\pi )}=r \\\\[-0.35em] ~\dotfill`

`\bf \textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}\qquad \qquad \begin{cases} r=(4)/(√(\pi ))\\ V=112 \end{cases}\implies 112=\cfrac{\pi \left( (4)/(√(\pi )) \right)^2(h)}{3} \\\\\\ 336=\pi \left( \cfrac{4^2}{(√(\pi ))^2} \right)h\implies 336=\pi \cdot \cfrac{16h}{\pi }\implies 336=16h \\\\\\ \cfrac{336}{16}=h\implies \blacktriangleright 21=h \blacktriangleleft`