Question

A conical vessel of radius 6 cm and height 8 cm is completely filled with water .A sphere is lowered into the water such that when it touches the sides, it is just immersed .What fraction of water overflows?

Answer 1

Answer:1

Step-by-step explanation:

Answer 2

Given:

Radius, PX

r = 6 cm

Height, PZ

h = 8 cm

To Find:

Fraction of overflowed water = ?

Solution:

According to the given values,

$PZ=\sqrt{(8)^2+(6)^2}$

⇒    $=\sqrt{64+36}$

⇒    $=10 \ cm$

Now,

$Volume = \frac{1}{3}\pi \ r^2h$

⇒           $=\frac{1}{3}\times \pi\times 6\times 6\times 8$

⇒           $=96\pi$

Therefore,

$Sin \theta=\frac{OM}{OZ} =\frac{PX}{PZ}$

            $\frac{h}{8-r} =\frac{6}{10}$

On applying cross-multiplication, we get

⇒          $10r=48-6r$

⇒          $10r+6r=48$

⇒           $16r=48$

⇒              $r=\frac{48}{16}$

⇒              $r=3$

Now,

Volume of the sphere = $\frac{4}{3}\pi \times 3\times 3\times 3$

                                     = $36\pi$

Fraction of overflowed water will be:

$=\frac{36\pi}{96\pi}$

$=3:8$