Question

A cyclindrical water tank has a radius of 6 feet and a height of 20 feet. A larger tank with the same height has a volume that is 3 times the volume of the smaller tank. What is the best approximation of the diameter of the larger tank?

Answer 1

We need to find the volume of the smaller cylinder.
`V= \pi r^2h`. For us, that looks like
`V= \pi (6)^2(20)`. So the volume is
`V=720 \pi`. 3 times that is the volume of the larger at
`V=2160 \pi`. Using that volume in the formula to solve for the radius we have
`2160 \pi = \pi (r^2)(20)`. Divide both sides by 20pi to get
`r^2=108`. Taking the square root of both sides gives us that r = 10.3923 or, in radical form, r =
`6 √(3)`. Diameter is twice that at 12sqrt(3)

Answer 2

`\bf \textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\ -----\\ r=6\\ h=20 \end{cases}\implies V=\pi 6^2(20)\implies \stackrel{\textit{smaller cylinder}}{V=720\pi }`

since the smaller tank has a volume of 720π, then the larger tank has a volume of 3 times that much or 3*720π, or 2160π, and its height is the same as the smaller one's, 20.


`\bf \textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\ -----\\ V=2160\pi \\ h=20 \end{cases}\implies 2160\pi =\pi r^2(20)\implies \cfrac{2160\pi }{20\pi }=r^2 \\\\\\ 108=r^2\implies √(108)=r\implies 6√(3)=r \\\\\\ \stackrel{\textit{and since the \underline{d}iameter is twice the radius}}{d = 2r\implies d=12√(3)}`