Related to surface area and volume of cone and sphere

Question

A conical vessel of radius 12 cm and height 16 cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed. What fraction of the water overflows?

Answer 1

The radius of sphere is r

R = 12 cm

h = 16 cm

Slant height = AO 

= \(\sqrt{16^2 + 12^2}\)

\(= 20\) cm

From, 

\(\triangle OBA \sim \triangle OCD\)

\(\frac{AB}{CD} = \frac{OA}{OD} \)

⇒ \(\frac{12}r = \frac{20}{16 - r}\)

⇒ \(192 - 12r = 20 r\)

⇒ \(r = 6 \) cm

V_water = \(\frac 13 \pi r^2 h\)

\(= \frac 13 \times \pi \times 12^2 \times 16 \)

\(= 768 \pi \) cm³

V_{water overflow} = \(\frac 43 \pi r^3\)

\(= \frac 43 \times \pi \times 6^3\)

\(= 288 \pi\) cm³

\(\cfrac{\text V_{\text{water overflow}}} {\text V_{\text{water}}}= \frac{288 \pi}{768 \pi} = \frac 38\)