Question

Rinku was very happy to receive a fancy jumbo pencil from his best friend Rohan on his birthday. Pencil is a basic writing tool, when sharpened its shape is a combination of cylinder & cone as given in the picture. Cylindrical pencil with conical head is a common shape worldwide since ages. Commonly pencils are made up of wood & plastic but we should promote pencils made up of eco-friendly material (many options available in the market these days) to save environment.

The dimensions of Rinku’s pencil are given as follows: 

Length of cylindrical portion is 21cm. Diameter of the base is 1 cm and height of the conical portion is 1.2 cm 

Based on the above information, answer the following questions:

(i) Find the slant height of the sharpened part.

(ii) Find curved surface area of sharpened part (in terms of π). 

(iii) (A) (B) Find the total surface area of the pencil (in terms of π). 

OR 

The pencil’s total height decreases by 8.2 cm after sharpening it many times, what is the volume of the cylindrical part of the shortened pencil (in terms of π)?

Answer 1

(i) l² = (1.2)² + (0.5)² 

= 1.44 + 0.25 

⇒ l = √1.69 = 1.3 

(ii) Curved surface area of sharpened part 

= π × 0.5 × 1.3 

= (0.65 π) cm² 

(iii) (A) Total surface area of pencil 

 = CSA of cylinder + CSA of cone + area of base circle 

 = π × 0.5 × 0.5 × 21 + 0.65 π + π × (0.5)² 

= (5.25 + 0.65 + 0.25)π 

= (6.15 π) cm² 

OR 

(B) Length of cylindrical part of shortened pencil 

= (21 − 8.2) cm 

= 12.8 cm 

 So, the volume of the cylindrical part of shortened pencil 

= π × 0.5 × 0.5 × 12.8 

= (3.2 π ) cm³