2. Problems based on conversion of solid from one shape to another

 

Solution:

 

Height of the cylinder \(=\) \(36 \ cm\)

 

The radius of the base of the cylinder \(=\) \(8 \ cm\)

 

Volume of the cylinder \(=\) \(\pi r^2 h\) cu. units

 

\(=\) \(\pi \times 8^2 \times 36\)

 

\(= 2304 \pi \ cm^3\)

 

Let '\(r\)' be the radius of the sphere.

 

Volume of sphere \(=\) \(\frac{4}{3} \pi r^3\) cu. units

 

Since the cylinder reshapes into a sphere, the volume remains unchanged.

 

Volume of sphere \(=\) Volume of the cylinder

 

\(\frac{4}{3} \pi r^3 = 2304 \pi\)

 

\(r^3\) \(=\) $\frac{2304\mathrm{\pi }\times 3}{\mathrm{\pi }\times 4}$

 

\(r^3\) \(=\) \(1728\)

 

\(r = 12\)

 

\(d = r \times 2 = 12 \times 2 = 24\)

 

Therefore, the diameter of the sphere is \(24 \ cm\).

 

 

2. How many spherical lead shots of radius \(3.5 \ cm\) can be made out of a solid rectangular lead piece with dimensions \(33 \ cm\), \(21 \ cm\) and \(14 \ cm\).

 

Solution:

 

The radius of the spherical lead shot \(=\) \(3.5 \ cm\)

 

Volume of spherical lead shot \(=\) \(\frac{4}{3} \pi r^3\) cu. units

 

\(=\) \(\frac{4}{3} \times \frac{22}{7} \times (3.5)^3\)

 

\(=\) \(\frac{539}{3}\) \(cm^3\)

 

Length of the rectangular lead piece \(=\) \(33 \ cm\)

 

\(=\) $\frac{9702\times 3}{539}$

 

\(=\) \(54\)

 

Therefore, \(54\) spherical leads can be made from the rectangular lead piece.