Hello, stones44!

A cone with a 10-inch diameter is 12 inches tall.

a. Find the radius of the largest sphere that can be inscribed in the cone.

b. The volume of this sphere is what percentage of the volume of the cone? Code:

- : - - - 5 - - - D - - - 5 - - - :
C * - - - - - - - * * * - - - - - - - * B
: \ * | * /
: \ * | * /
: \ * |r * /
: \ | /
: \ * | * /
: \ * O* * /
: \* | \ */
12 \ | \r /
: * | \ *
: * | *E
: \* | */
: \ * * * /
: \ | /
: \ | /
: \ | /
: \ | /
: \|/
- *
A

We have: .$\displaystyle DB = 5,\:DA = 12$

. . From Pythagorus: .$\displaystyle DA = 13$

The center of the sphere is $\displaystyle O.$

Radii $\displaystyle OD = OE = r$

. . Hence: .$\displaystyle PA = 12-r$

We have similar right triangles: .$\displaystyle \Delta OEA \sim \Delta BDA$

So we have: .$\displaystyle \frac{OE}{OA} \:=\:\frac{BD}{AB}\quad\Rightarrow\quad\frac{r}{12-r} \:=\:\frac{5}{13}$

Therefore: .$\displaystyle \boxed{r \:=\:\frac{10}{3}}$

.