The surface area of a sphere with a radius of $5 \, cm$ is five times the curved surface area of a cone with a radius of $4 \, cm$. Find the height and volume of the cone (take $\pi = \frac{22}{7}$).

(N/A) Surface area of the sphere $= 4\pi r^2 = 4 \times \pi \times 5^2 = 100\pi \, cm^2$.
Curved surface area of the cone $= \pi rl = \pi \times 4 \times l = 4\pi l \, cm^2$,where $l$ is the slant height of the cone.
According to the problem,the surface area of the sphere $= 5 \times$ curved surface area of the cone.
$100\pi = 5 \times 4\pi l$
$100\pi = 20\pi l$
$l = \frac{100}{20} = 5 \, cm$.
To find the height $h$ of the cone,use the formula $l^2 = h^2 + r^2$:
$5^2 = h^2 + 4^2$
$25 = h^2 + 16$
$h^2 = 25 - 16 = 9$
$h = 3 \, cm$.
Volume of the cone $= \frac{1}{3} \pi r^2 h = \frac{1}{3} \times \frac{22}{7} \times 4^2 \times 3$
$= \frac{22}{7} \times 16 = \frac{352}{7} \approx 50.29 \, cm^3$.