Name: Conversion of Solids
Uploaded: 2019-07-19
Duration: 5 min 33 s
Description: What happens when a goldsmith heats a gold bar to create something? The gold bar's shape gets changed. But what about its volume? Does it change too? Find ou...

Question

An open cylindrical vessel of intenal diameter

7 cm

and height

8 cm

stands on a horizontal table. Inside this is placed a sohd metallic right circular cone, the diameter of whose base is

3 \frac{1}{2} cm

and height

8 cm

. Find the volume of water required to fill the vessel. If this cone is replaced by another cone, whose height is

1 \frac{3}{4} cm

and the radius of whose base is

2 cm

,find the drop in the water level.

(π = \frac{22}{7})

Accepted Answer

Diameter of base of cylinder $= 7 cm$

Therefore,radius of cylinder $= \frac{7}{2} cm$

Volume of cylinder $= {πr}^{2} h$

$= \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times 8 = 308 {cm}^{3}$

Diameter of the base of cone $= \frac{7}{2} cm$

Volume of cylinder $= {πr}^{2} h$

Therefore, radius of cone $= \frac{7}{4} cm$

Volume of cone $= \frac{1}{3} {πr}^{2} h$

$= \frac{1}{3} \times \frac{22}{7} \times \frac{7}{4} \times \frac{7}{4} \times 8 = \frac{77}{3} {cm}^{3}$

On placing the cone into cylindrical vessel, the volume of remaining portion where the water is to be filled

$= 308 - \frac{77}{3}$

$= \frac{924 - 77}{3}$

$= \frac{847}{3}$

$= 282.33 {cm}^{3}$

Height of new cone $= 1 \frac{3}{4} = \frac{7}{4} cm$

Radius $= 2 cm$

Therefore, volume of new cone

$= \frac{1}{3} {πr}^{2} h = \frac{1}{3} \times \frac{22}{7} \times 2 \times 2 \times \frac{7}{4} = \frac{22}{3} {cm}^{3}$ Volume of water which comes down

$= \frac{77}{3} - \frac{22}{3} {cm}^{3} = \frac{55}{3} {cm}^{3} . . . . . (i)$

Let $h$ be the height of water which is dropped down.

Radius $= \frac{7}{2} cm$

$∴$ Volume $= {πr}^{2} h = \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times h = \frac{77}{2} h . . . . . (ii)$ From equation $(i)$ and $(ii)$

$= \frac{77}{2} h = \frac{55}{3}$

$\Rightarrow h = \frac{55}{3} \times \frac{2}{77}$

$\Rightarrow h = \frac{10}{21} cm$

$∴$ Drop in water level $= \frac{10}{21} cm$

Important Questions on Cylinder, Cone and Sphere (Surface Area and Volume)

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