Question

On the day of her examination, Riya sharpened her pencil from both ends as shown below.

The diameter of the cylindrical and conical part of the pencil is 4.2 mm. If the height of each conical part is 2.8 mm and the length of the entire pencil is 105.6 mm, find the total surface area of the pencil.

Hint:

Total surface area = curved surface of cylinder + curved surface area of two cones. Use Pythagoras to find slant height of cone.

Answer 1

Given:
- Diameter of pencil (cylindrical and conical parts) = 4.2 mm
- Height of each conical part = 2.8 mm
- Length of entire pencil = 105.6 mm

Step 1: Calculate radius of pencil
\[ r = \frac{4.2}{2} = 2.1 \, \text{mm} \]

Step 2: Calculate height of cylindrical part
Length of cylindrical part = Total length - height of two cones
\[ h_{\text{cyl}} = 105.6 - 2 \times 2.8 = 105.6 - 5.6 = 100 \, \text{mm} \]

Step 3: Calculate surface area of cylindrical part
Curved surface area (CSA) of cylinder = \(2 \pi r h\)
\[ = 2 \times \frac{22}{7} \times 2.1 \times 100 = 1320 \, \text{mm}^2 \]

Step 4: Calculate slant height of conical part
\[ l = \sqrt{r^2 + h_{\text{cone}}^2} = \sqrt{(2.1)^2 + (2.8)^2} = \sqrt{4.41 + 7.84} = \sqrt{12.25} = 3.5 \, \text{mm} \]

Step 5: Calculate surface area of two cones
Lateral surface area (LSA) of one cone = \(\pi r l\)
\[ = \frac{22}{7} \times 2.1 \times 3.5 = 33 \, \text{mm}^2 \] For two cones:
\[ 2 \times 33 = 66 \, \text{mm}^2 \]

Step 6: Calculate total surface area of pencil
\[ \text{Total surface area} = \text{CSA of cylinder} + \text{LSA of two cones} = 1320 + 66 = 1386 \, \text{mm}^2 \]

Final Answer:
Total surface area of the pencil = \(\boxed{1386 \, \text{mm}^2}\)