Question

Singing bowls (hemispherical in shape) are commonly used in sound healing practices. Mallet (cylindrical in shape) is used to strike the bowl in a sequence to produce sound and vibration. In the given figure, the hemispherical bowl has outer radius 6 cm and inner radius 5 cm, and the mallet has a height of 10 cm and radius 2 cm.

Based on the above information, answer the following questions. [Use π = 3.14]

1. What is the volume of the material used for making the mallet?
2. The bowl is to be polished from inside. Find the inner surface area of the bowl.
3. Find the volume of metal used to make the bowl.
4. Find the total surface area of the mallet.

Answer 1

i. Volume of the material used for making the mallet

Given: Radius of the mallet, \[r_c = 2\text{ cm}\]

Height of the mallet, \[h = 10\text{ cm}\]

Formula: \[\text{Volume} = \pi r_c^2 h\]

Calculation: \[\text{Volume} = 3.14 \times (2)^2 \times 10\]

\[\text{Volume} = 3.14 \times 4 \times 10 = 125.6\text{ cm}^3\]

ii. Inner surface area of the bowl

Given: Inner radius of the hemispherical bowl, \[r = 5\text{ cm}\]

Formula: \[\text{Inner Surface Area} = 2\pi r^2\]

Calculation: \[\text{Inner Surface Area} = 2 \times 3.14 \times 5^2\]

\[\text{Inner Surface Area} = 2 \times 3.14 \times 25 = 157\text{ cm}^2\]

iii. Volume of metal used to make the bowl

Given: Outer radius, \[R = 6\text{ cm}\]

Inner radius, \[r = 5\text{ cm}\]

Formula: \[\text{Volume} = \frac{2}{3}\pi (R^3 - r^3)\]

Calculation: \[\text{Volume} = \frac{2}{3} \times 3.14 \times (6^3 - 5^3)\]

\[\text{Volume} = \frac{2}{3} \times 3.14 \times (216 - 125)\]

\[\text{Volume} = \frac{2}{3} \times 3.14 \times 91 \approx 190.49\text{ cm}^3\]

iv. Total surface area of the mallet

Formula: \[\text{TSA} = 2\pi r_c (h + r_c)\]

Calculation: \[\text{TSA} = 2 \times 3.14 \times 2 \times (10 + 2)\]

\[\text{TSA} = 12.56 \times 12 = 150.72\text{ cm}^2\]