Two way tube

Geometry Level pending

A rectangular piece of paper can be rolled into a tube in two different ways. (Edges should not overlap.)

When it is rolled up the long way the volume enclosed is $\frac{2500}{\pi} cm^{3}$

When it is rolled up the short way the volume enclosed is $\frac{3125}{\pi} cm^{3}$

Find the area of the piece of paper in $cm^{2}$

1 solution

Jeremy Galvagni
Apr 13, 2018

Call the dimensions $x$ and $y$ .

It is simple enough to show the volumes are $\frac{x^{2}y}{4\pi}$ and $\frac{xy^{2}}{4\pi}$

This gives $x^{2}y=10000$ and $xy^{2}=12500$

Solve the first equation for $y$ and substitute into the second

$x(\frac{10000}{x^{2}})^{2}=12500$

$x^{3}=\frac{10000^{2}}{12500}$

$x=20$

Similarly $y=25$

$xy = 20*25 = \boxed{500}$