A hemisphere dome of a building needs to be painted. If the circumference of the base of the dome is 17.6 m, then find the cost of painting it, given that the cost of painting is ₹5 per 100 cm².

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Points to Remember:

  • The problem involves calculating the surface area of a hemisphere.
  • Unit conversion is crucial (meters to centimeters).
  • The cost calculation requires multiplying the surface area by the cost per unit area.

Introduction:

This question requires a factual and analytical approach to determine the cost of painting a hemispherical dome. We are given the circumference of the base of the dome and the cost of painting per 100 cm². To find the total cost, we first need to calculate the surface area of the hemisphere and then convert the units to ensure consistency before performing the cost calculation. The formula for the surface area of a sphere is 4πr², where ‘r’ is the radius. Since we are dealing with a hemisphere, we will use half of this value.

Body:

1. Calculating the Radius:

The circumference of the base (which is a circle) is given as 17.6 meters. The formula for the circumference of a circle is 2πr. Therefore:

2πr = 17.6 m

r = 17.6 m / (2π) ≈ 2.8 m

Converting meters to centimeters:

r ≈ 2.8 m * 100 cm/m = 280 cm

2. Calculating the Surface Area:

The surface area of a hemisphere (excluding the base) is given by 2πr². Substituting the value of ‘r’:

Surface area = 2π(280 cm)² ≈ 492600 cm²

3. Calculating the Total Cost:

The cost of painting is ₹5 per 100 cm². To find the total cost, we divide the total surface area by 100 and then multiply by the cost per 100 cm²:

Total cost = (492600 cm² / 100 cm²) * ₹5 = ₹24630

Conclusion:

In summary, the total cost of painting the hemispherical dome is ₹24630. This calculation involved determining the radius from the given circumference, calculating the surface area of the hemisphere, and then using the provided cost per unit area to arrive at the final cost. This approach ensures accuracy and clarity in solving the problem. While this calculation is straightforward, it highlights the importance of accurate measurements and unit conversions in practical applications, ensuring efficient resource allocation in projects like building maintenance and painting. Future projects should emphasize precise measurements and detailed cost estimations to avoid unexpected expenses and ensure project success.

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