Ellipse
Ellipse on paper rolled into cylinder — volume
MJAT_TS4_P2
Grade 12
Question:
The ellipse $\dfrac{x^2}{100}+\dfrac{y^2}{19}=1$ is printed on a $20\times 20$ sheet of paper so that the ellipse is centred and its axes are parallel to the edges. The sheet is then rolled up into a cylinder so that the two foci of the ellipse are physically touching one other. If $V$ is the volume of this cylinder, then $\dfrac{V}{\pi}=$
Step-by-Step Solution
Key Concept: For the ellipse $x^2/100+y^2/19=1$: $a^2=100$, $b^2=19$, $c^2=81$, $c=9$. Foci at $(\pm 9,0)$, distance apart $=18$. When rolling along the $x$-direction: circumference $=20$ (sheet width), so foci touching means the sheet is rolled so their separation = circumference: $2c=2\pi R\Rightarrow R=9/\pi$. Height $=20$.
$V=\pi R^2 h=\pi(9/\pi)^2\times 20=\pi\times 81/\pi^2\times 20=1620/\pi$. $V/\pi=\mathbf{1620}$.
Correct Answer: 1620