Question

Find the number of solid spheres, each of diameter $6{\text{ cm}}$, that could be moulded to form a solid metallic cylinder of height ${\text{45 cm}}$ and diameter ${\text{4 cm}}$.

Answer 1

Hint:In order to find the number of solid spheres that can be taken out from a metallic cylinder we need to find the volume of the cylinder, and then find the volume of the spheres. And then, dividing the volume of the cylinder by the volume of the sphere and the value obtained is the number of spheres.

Formula used:
Volume of Cylinder: = $Volume\left( V \right) = \pi {R^2}H$
Volume of Sphere: = $Volume\left( v \right) = \dfrac{4}{3}\pi {r^3}$

Complete step by step answer:
We are given the dimensions of a solid metallic cylinder and the dimensions of the solid spheres.Writing out the dimensions of the solid metallic cylinder:
Height (H) ${\text{ = 45 cm}}$
Diameter ${\text{ = 4 cm}}$
Since, we know that radius is equal to $\dfrac{{diameter}}{2}$.
So, we get $radius\left( R \right) = \dfrac{4}{2} = 2{\text{ cm}}$.
From the formula for volume of cylinder, we know that:
$Volume\left( V \right) = \pi {R^2}H$
So, substituting the values in the above formula, we get:
$ \Rightarrow Volume\left( V \right) = \pi {\left( 2 \right)^2} \times 45$
$ \Rightarrow Volume\left( V \right) = \pi \times 4 \times 45$
$ \Rightarrow Volume\left( V \right) = 180\pi $ …….(1)
Therefore, the Volume of the cylinder is equal to $180\pi $.

Writing the dimension of the solid metallic sphere:
Diameter ${\text{ = 6 cm}}$
Since, we know that radius is equal to $\dfrac{{diameter}}{2}$.
So, we get $radius\left( r \right) = \dfrac{6}{2} = 3{\text{ cm}}$.
From the formula for volume of sphere, we know that:
$Volume\left( v \right) = \dfrac{4}{3}\pi {r^3}$
So, substituting the values in the above formula, we get:
$ \Rightarrow Volume\left( v \right) = \dfrac{4}{3}\pi {\left( 3 \right)^3}$
$ \Rightarrow Volume\left( V \right) = \pi \times \dfrac{4}{3} \times 27$
$ \Rightarrow Volume\left( V \right) = \pi \times 4 \times 9$
$ \Rightarrow Volume\left( V \right) = 36\pi $ …….(2)
Therefore, the Volume of the cylinder is equal to $36\pi $.

Now, to find the number of spheres that can be taken out from the cylinder, we divide the volume of the cylinder by the volume of the sphere.
So, dividing the equation 1 by equation 2, we get:
$ \Rightarrow {\text{number of spheres = }}\dfrac{V}{v}$
Substituting the value of equation 1 and equation 2:
$ \Rightarrow {\text{number of spheres = }}\dfrac{{180\pi }}{{36\pi }}$
$ \therefore {\text{number of spheres = 5}}$

Therefore, five (5) spheres can be taken out from the cylinder.

Note:We have used volume because the shapes are solid and completely filled.For any other shapes we would use the same way for example taking out cylinders from a solid sphere, or cones from cylinders, etc. by taking out volume and dividing them.