Question

A Ramraj pencil box of dimensions 30cm × 5cm × 5cm consists of 10 sharpened cylindrical pencils placed in a vertical fashion each of length 20cm and diameter 1.4cm and having a conical surmount whose slant height is 2.5m. There are 2 rows of 5 pencils each. Find available volume in the box. Can a sharpener and an eraser in shape of a cube each of side 5cm be placed on top of the pencils such that sharpener is placed on top of the eraser.

• A. 429.8 , No
• B. 426.8 , Yes
• C. 426.5 , No
• D. 426.5 , Yes

Answer 1

The correct option is A

429.8 , No

Lets compute the volume of a sharpened pencil.

Volume of 1 pencil = volume of the cylindrical part + volume of the conical part

= $\pi r^{2}H+\frac{1}{3}\pi r^{2}h$
$h^2=(2.5{)}^2-(0.7{)}^2=5.76$
$\Rightarrow h=2.4$

= $3.14\times {0.7}^2[20+\frac{2.4}{3}]$

= 32.02 $c{m}^3$

Volume of 10 pencils =320.2 $c{m}^3$

Volume of the pencil box = $30\times 5\times 5$ = 750 $c{m}^3$

Available volume = 750 – 320.2 = 429.8 $c{m}^3$

Analysing the volume, the answer should be a yes, but considering the dimensions of the box we cannot have both an eraser and a sharpener.

Length of the box=30cm.

Length of the sharpened pencil = 20 + 2.4 = 22.4

Length of the eraser and sharpener = 10cm

Since length of the sharpener +eraser+pencil > length of box , it is not possible.