Question

Two spheres of metal weigh 1 kg and 7kg . The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere.

Answer 1

Hint: Use the radius of the first metal sphere to find the radius of the other sphere as their densities are the same. Use the fact that the big sphere is made up of smaller spheres thus the sum of their volumes would be equal to the volume of the big sphere to get to the desired answer.

Complete step-by-step answer:

We know that $density = $ $\dfrac{{mass}}{{volume}}$
It is given that the density of both spheres is the same as they are made of the same metal . Since both the spheres are made of the same metal, their weights are directly proportional to their volumes, or the ratio of their volumes is equal to the ratio of their weights.
Radius of the first small sphere, ${r_1}$ = 3cm.
Let the radius of second small sphere be ${r_2}$ cm
As the weights are directly proportional to the volumes of both the sphere , we get
$\dfrac{{{v_1}}}{{{v_2}}} = \dfrac{1}{7}$ ( where ${v_1}$ and ${v_2}$ are the volumes of the first and the second sphere respectively )
Now we know that the volume of the sphere is $\dfrac{4}{3}\pi {r^3}$
Therefore,
$\dfrac{{\dfrac{4}{3}\pi {r_1}^3}}{{\dfrac{4}{3}\pi {r_2}^3}} = \dfrac{1}{7} \Rightarrow \dfrac{{{3^3}}}{{{r_2}^3}} = \dfrac{1}{7}$ ( where ${r_1}$and ${r_2}$ are the radii of both spheres )
$ \Rightarrow {r_2}^3 = 27 \times 7 = 189$
Sum of volumes of both sphere =$\dfrac{4}{3}\pi \left( {{r_1}^3 + {r_2}^3} \right) = \dfrac{4}{3}\pi \left( {{3^3} + 189} \right) = \dfrac{4}{3}\pi \times 216 c{m^3}$
Let the radius of the big sphere be R
Volume of the big sphere = $\dfrac{4}{3}\pi {R^3}$
Since the big sphere is made up of both the small spheres , the total volume of both the spheres is equal to the volume of the big sphere .
$ \Rightarrow \dfrac{4}{3}\pi {R^3} = \dfrac{4}{3}\pi \times 216$
$ \Rightarrow {R^3} = 216 c{m^3} \Rightarrow R = 6$ cm
Therefore diameter = 2R = 12 cm

Note: In such types of questions we need to understand the concept related to volume and density of spheres and recall all the formulas. Remember that if the densities are the same the ratio of volumes is equal to the ratio of their weights of any two objects.