Question

1. a sphere is inscribed in a cone of radius 5 and height 12. find the radius of the sphere.

Explanation:

Step1: Find slant height of cone

Slant height $l = \sqrt{5^2 + 12^2} = 13$

Step2: Set up similar triangles

Let sphere radius = $r$. The smaller triangle (from center of sphere to slant height) has legs $r$ and $12 - r$, and the cone's triangle has legs 5 and 12. So $\frac{r}{5} = \frac{12 - r}{13}$

Step3: Solve for $r$

$13r = 5(12 - r) \implies 13r = 60 - 5r \implies 18r = 60 \implies r = \frac{60}{18} = \frac{10}{3}$

Answer:

$\frac{10}{3}$