Question

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

Hint:

When two solids are joined, omit the common base area and add only the curved surfaces for total surface area calculations.

Answer 1

Step 1: Given data.
Radius \( r = 3.5 \text{ cm} \)
Total height of the toy \( = 15.5 \text{ cm} \)
Height of cone \( h = 15.5 - 3.5 = 12 \text{ cm} \).

Step 2: Formula for total surface area (TSA).
\[ \text{TSA} = \text{Curved surface area of cone} + \text{Curved surface area of hemisphere} \]
Step 3: Calculate slant height of cone.
\[ l = \sqrt{r^2 + h^2} = \sqrt{(3.5)^2 + (12)^2} = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5 \text{ cm} \]
Step 4: Calculate each area.
Curved surface area of cone \( = \pi r l = \pi (3.5)(12.5) = 43.75\pi \text{ cm}^2 \).
Curved surface area of hemisphere \( = 2\pi r^2 = 2\pi (3.5)^2 = 2\pi (12.25) = 24.5\pi \text{ cm}^2 \).
Step 5: Total surface area.
\[ \text{TSA} = 43.75\pi + 24.5\pi = 68.25\pi \] \[ \text{TSA} = 68.25 \times 3.14 = 214.3 \text{ cm}^2 \] Step 6: Conclusion.
Hence, the total surface area of the toy is \( \boxed{214.3\ \text{cm}^2} \).