Calculating Painting Cost for Hemispherical Bowl

This problem involves calculating the total surface area to be painted for a hemispherical bowl and then determining the cost based on the given rate.

Understanding the Problem

• We have a hemispherical bowl with a given radius.
• The bowl needs to be painted inside and outside.
• The thickness of the bowl is negligible, meaning the inner and outer curved surfaces have the same radius.
• We need to find the total cost of painting at a specified rate per square centimeter.

Key Information

• Radius of the hemispherical bowl, r = 21 cm
• Area to be painted: Inner curved surface area + Outer curved surface area
• Cost of painting = ₹0.05 per $cm^2$
• Value of $\pi = \frac{22}{7}$

Formulas Used

The curved surface area of a hemisphere is given by the formula:

$$ A_{curved\_hemisphere} = 2 \pi r^2 $$

Where:

• A is the curved surface area
• r is the radius
• $\pi$ is the mathematical constant Pi

Step-by-Step Solution

1. Calculate the Inner Surface Area

Since the thickness is negligible, the inner surface area is the curved surface area of the hemisphere:

$$ A_{inner} = 2 \pi r^2 $$

Substitute the values:

$$ A_{inner} = 2 \times \frac{22}{7} \times (21 \text{ cm})^2 $$

$$ A_{inner} = 2 \times \frac{22}{7} \times 21 \times 21 \text{ cm}^2 $$

$$ A_{inner} = 2 \times 22 \times 3 \times 21 \text{ cm}^2 \quad (\text{since } \frac{21}{7} = 3) $$

$$ A_{inner} = 44 \times 63 \text{ cm}^2 $$

$$ A_{inner} = 2772 \text{ cm}^2 $$

2. Calculate the Outer Surface Area

Because the thickness is negligible, the outer surface area is the same as the inner surface area:

$$ A_{outer} = 2 \pi r^2 = 2772 \text{ cm}^2 $$

3. Calculate the Total Area to be Painted

The total area includes both the inside and the outside surfaces:

$$ A_{total} = A_{inner} + A_{outer} $$

$$ A_{total} = 2772 \text{ cm}^2 + 2772 \text{ cm}^2 $$

$$ A_{total} = 5544 \text{ cm}^2 $$

4. Calculate the Total Cost of Painting

The cost is ₹0.05 per $cm^2$. To find the total cost, multiply the total area by the rate:

$$ \text{Total Cost} = A_{total} \times \text{Rate per } cm^2 $$

$$ \text{Total Cost} = 5544 \text{ cm}^2 \times ₹0.05 / cm^2 $$

$$ \text{Total Cost} = 5544 \times \frac{5}{100} \text{ Rupees} $$

$$ \text{Total Cost} = 5544 \times \frac{1}{20} \text{ Rupees} $$

$$ \text{Total Cost} = \frac{5544}{20} \text{ Rupees} $$

$$ \text{Total Cost} = \frac{2772}{10} \text{ Rupees} $$

$$ \text{Total Cost} = ₹277.20 $$

Conclusion

The total cost of painting the hemispherical bowl inside and outside is ₹277.20.