Question

Rasheed got a spinning top (Lattu) as his birthday present. The top is shaped like a cone surmounted by a hemisphere. The entire top is 5 cm in height and the diameter of the top is 3.5 cm as shown in the figure. Find the area Rasheed has to colour. (Take } $\pi = \dfrac{22}{7}$ 

Hint:

For combined solids, always find curved surface areas (not bases) and add them together for total colouring or painting.

Answer 1

Step 1: Given data.
Diameter of the top = 3.5 cm \[ \Rightarrow \text{Radius (r)} = \dfrac{3.5}{2} = 1.75 \, \text{cm} \] Total height of the top = 5 cm
The height of cone \( h_1 = 5 - 1.75 = 3.25 \, \text{cm} \) (subtracting hemisphere radius).

Step 2: Find the slant height of cone.
\[ l = \sqrt{r^2 + h_1^2} = \sqrt{(1.75)^2 + (3.25)^2} = \sqrt{3.06 + 10.56} = \sqrt{13.62} = 3.69 \, \text{cm} \]
Step 3: Curved surface area (C.S.A.) of cone.
\[ \text{C.S.A. of cone} = \pi r l = \dfrac{22}{7} \times 1.75 \times 3.69 = 22 \times 0.25 \times 3.69 = 14.19 \, \text{cm}^2 \]
Step 4: Curved surface area of hemisphere.
\[ \text{C.S.A. of hemisphere} = 2\pi r^2 = 2 \times \dfrac{22}{7} \times (1.75)^2 = 2 \times \dfrac{22}{7} \times 3.06 = 19.2 \, \text{cm}^2 \] Step 5: Total surface area to be coloured.
\[ \text{Total area} = 14.19 + 19.2 = 33.39 \, \text{cm}^2 \] (Considering round-off and possible figure scale, total approximate area = 33.4 cm$^2$.)
Step 6: Conclusion.
Hence, Rasheed has to colour approximately 33.4 cm$^2$ of the surface.