Twelve solid spheres of the same size

Question:

Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 2 cm and height 16 cm. The diameter of each

sphere is

(a) 4 cm                     

(b) 3 cm                    

(c) 2 cm                    

(d) 6 cm

Solution:

(c) Given, diameter of the cylinder = 2 cm

∴ Radius = 1 cm and height of the cylinder = 16 cm                    [∵ diameter = 2 x radius]

∴ Volume of the cylinder = π x (1)2 x 16 = 16 π cm3

$\left[\because\right.$ volume of cylinder $=\pi \times(\text { radius })^{2} \times$ height $]$

Now, let the radius of solid sphere $=r \mathrm{~cm}$

Then, its volume $=\frac{4}{3} \pi r^{3} \mathrm{~cm}^{3} \quad\left[\because\right.$ volume of sphere $\left.=\frac{4}{3} \times \pi \times(\text { radius })^{3}\right]$

According to the question,

 

Volume of the twelve solid sphere = Volume of cylinder

$\Rightarrow \quad 12 \times \frac{4}{3} \pi r^{3}=16 \pi$

$\Rightarrow \quad r^{3}=1 \Rightarrow r=1 \mathrm{~cm}$

∴                    Diameter of each sphere, d=2r = 2×1=2 cm

Hence, the required diameter of each sphere is 2 cm.

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