Question

The triangle $ABC$ has angle $B={90}^{\circ }$. When it is rotated about $AB$, it gives a cone of volume $800\pi$. When it is rotated about $BC$, it gives a cone of volume $1920\pi$. The length of $AC$ is

Answer 1

When rotated about $AB$,
$\frac{1}{3}\pi x^{2}y=800\pi \cdots \left(1\right)$

When rotated about $BC$,
$\frac{1}{3}\pi y^{2}x=1920\pi \cdots \left(2\right)$

Dividing $\left(2\right)$ by $\left(1\right)$, we get
$\frac{y}{x}=\frac{192}{80}=\frac{12}{5}$
$\Rightarrow y=\frac{12}{5}x\cdots \left(3\right)$

Multiplying $\left(1\right)$ and $\left(2\right)$, we get
$\frac{1}{9}{\pi }^2{x}^3{y}^3={10}^3\cdot 2^3\cdot 192{\pi }^2$
$\Rightarrow x^3{y}^3={10}^3\cdot 2^3\cdot 3^2\cdot 3\cdot 64={10}^3\cdot 2^3\cdot 3^3\cdot 4^3$
$\Rightarrow xy=10\cdot 2\cdot 3\cdot 4=240$
From $\left(3\right)$, $x\cdot \frac{12}{5}x=240$
$\Rightarrow x^2=20\times 5=100$
$\Rightarrow x=10$
$\therefore y=\frac{12}{5}\times 10=24$
$\therefore AC=\sqrt{x^2+y^2}=\sqrt{{10}^2+{24}^2}=26$