Question

A cylindrical pencil is sharpened to produce a perfect cone at one end with no over all loss of its length. The diameter of the pencil is 1cm and the length of the conical portion is 2cm. Calculate the volume of the shavings. Give your answer correct to two places if it is in decimal [use π = 355/113].

Answer 1

The volume of the shavings will be the difference between the end of the pencil and the cone. Therefore,

VShavings = VPencil - VCone = πr2h - (πr2h)/3 = (3πr2h)/3 - (πr2h)/3 = (2πr2h)/3 = (2π(0.5cm)2(2cm))/3 = 1.0472cm3

*Note: r = radius, and the diameter is twice to radius. Thus, r = d/2 = 1cm/2 = 0.5cm

Answer 2

Given,

Diameter of the pencil = 1cm

The length of the conical portion is 2cm.

To find out,

Calculate the volume of the shavings.

Solution:

Radius of the pencil is 0.5cm.

Length of the conical portion( h) = 2cm

Volume of peels = = Volume of cylinder of length 2CM and base radius 0.5cm - Volume of the cone formed by this cylinder.

$Volume \: of \: peels \: \\ \\ = \pi \: {r}^{2} h - \frac{1}{3} \pi \: {r}^{2}h \\ \\ = \frac{2}{3} \pi \: {r}^{2} h \\ \\ = \frac{2}{3} \times \frac{355}{113} \times 0.5 \times 0.5 \times 2 \\ \\ = 1.05 \: {cm}^{3}$

Therefore the volume of the peels is 1.05 cubic units.