Question

What is the number of spherical balls of 2.5 mm diameter that can be obtained by melting a semicircular disc of 8 cm diameter and 2 cm thickness?
A.6144
B.3072
C.1536
D.768

Answer 1

Hint: To find the number of spherical ball can be obtained from a semi circular disc can be calculated as first calculate the volume of on spherical ball then calculate the volume of the semicircular disc and last to calculate number of balls divide the volume of semicircular disc by the volume of the ball. We will get the number of balls.

Complete step-by-step answer:
Given the diameter of spherical ball is 2.5 mm
Then Radius \[ = \dfrac{8}{2} = 4cm = \dfrac{{1.25}}{{10}}cm\]
We know the volume of sphere is \[\dfrac{4}{3}\pi {r^3}\]
On putting the given value of radius we get,
Then volume of the spherical ball \[ = \dfrac{4}{3}\pi {\left( {\dfrac{{1.25}}{{10}}} \right)^3}c{m^3}\]
And given diameter of semicircular disc is 8 cm and thickness 2 cm
Then radius \[ = \dfrac{8}{2} = 4cm\] and its height is 2cm
We will use the formula of volume of cylinder and divide it with 2 such that we will get the volume of the semicircular disc
We know the volume of a cylinder is \[\pi {r^2}h\]
On putting the given values we get,
Then volume of semicircular disc \[ = \dfrac{1}{2}\pi {\left( 4 \right)^2} \times 2 = 16\pi c{m^3}\]
 $ \Rightarrow $ Then number of balls = $\dfrac{\text{volume of disc}}{\text{ volume of ball}}$
\[ = \dfrac{{16\pi c{m^3}}}{{\dfrac{4}{3}\pi \left( {\dfrac{{1.25}}{{10}}} \right){^3}c{m^3}}} \\
   = \dfrac{{16 \times 3 \times 1000}}{{4 \times 1.953}} \\
   = 6144 \\
\]
​Hence the total number of balls that can be form from the semi circular disc is 6144

So, the correct answer is “Option A”.

Note: Students are aware that here semicircular disc behaves like a semi cylinder hence the volume will be half of the volume of the cylinder and using the formula of the same else number of semi circular disc obtained will be wrong.