Q.11. A hemispherical dome needs to be painted. If the circumference of the base of the dome is 154 cm , then find the cost of painting it if the cost of painting is \( ₹ 4 \) per \( 100 \mathrm{~cm}^{2} \) (use \( \pi=\frac{22}{7} \) ).

To find the cost of painting the hemispherical dome, we first need to determine its surface area. The formula for the surface area of a hemisphere (excluding the base) is \( 2\pi r^2 \). Given that the circumference of the base is \( 154 \, \text{cm} \), we can find the radius \( r \): \[ C = 2\pi r \implies r = \frac{C}{2\pi} = \frac{154}{2 \times \frac{22}{7}} = 7 \text{ cm}. \] Now, substituting \( r \) into the surface area formula: \[ \text{Surface Area} = 2\pi r^2 = 2 \times \frac{22}{7} \times (7^2) = 2 \times \frac{22}{7} \times 49 = 2 \times \frac{1078}{7} = \frac{2156}{7} \approx 308 \, \text{cm}^2. \] The cost to paint the dome: \[ \text{Cost} = \frac{308}{100} \times 4 = ₹ 12.32. \] Thus, the total cost of painting the hemispherical dome is approximately \( ₹ 12.32 \). The total surface area to paint is crucial in many real-world applications, such as construction and architecture, where accurate measurements can save resources and provide cost estimates. In this case, knowing the precise dimensions helped to avoid wasting paint and ensured a clean and professional finish on the dome. When tackling geometry problems like this one, a common mistake is misapplying the formulas or forgetting to convert units properly. Always double-check calculations and ensure you're using the right parameters—like substituting \( \pi \) correctly! Keeping track of units is equally vital to avoid costly errors in practical applications.