Question

A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of the balls are 1.5 cm and 2 cm. Find the diameter of the third ball.

Answer 1

Hint: In this particular type of question use the concept that volume of the original sphere is equal to the sum of the volume of the three small spheres, where as volume of sphere is given as, $4\pi {r^2}$cubic units, so use these concepts to reach the solution of the question.

Complete step-by-step answer:

Given data:
Radius of the original big ball is 3 cm.
Therefore, r = 3 cm, where r is the radius of the big ball.
Now this ball is melted and cast into three spherical balls.
Let the radius of these three spherical balls be ${r_1},{r_2},{r_3}$cm respectively.
Now it is given that the radii of two of the balls are 1.5 cm and 2 cm.
Therefore, ${r_1} = 1.5$ cm and ${r_2} = 2$ cm.
Now as we know that the volume of the sphere = $4\pi {r^2}$cubic units, where r is the radius of the sphere.
So the volume of the original sphere is equal to the sum of the volume of the three small spheres.
Therefore,
$ \Rightarrow 4\pi {r^2} = 4\pi r_1^2 + 4\pi r_2^2 + 4\pi r_3^2$
Now cancel out $4\pi $from both the sides in the above equation we have,
$ \Rightarrow {r^2} = r_1^2 + r_2^2 + r_3^2$
Now substitute the values in the above equation we have,
$ \Rightarrow {3^2} = {\left( {1.5} \right)^2} + {\left( 2 \right)^2} + r_3^2$
$ \Rightarrow {3^2} - {\left( {1.5} \right)^2} - {\left( 2 \right)^2} = r_3^2$
$ \Rightarrow 9 - 2.25 - 4 = r_3^2$
$ \Rightarrow r_3^2 = 9 - 6.25 = 2.75$
Now take square root on both sides we have,
$ \Rightarrow \sqrt {r_3^2 = } \sqrt {2.75} $
$ \Rightarrow {r_3} = 1.658$ cm.
Now as we know that the diameter is double of the radius.
So the diameter of the third sphere is,
$ \Rightarrow {d_3} = 2\left( {{r_3}} \right) = 2\left( {1.658} \right) = 3.316$ cm.
So this is the required diameter of the third ball.

Note: Whenever we face such types of questions the key concept is the volume of the sphere which remains constant, whether we cast the big sphere into 3 balls or into more than three balls the volume always remain constant an always remember that the diameter is twice of the radius, so first find out the radius of the third sphere as above then multiplied by 2 we will get the required diameter of the third sphere.