Question

A dome of a building is in the form of a hemisphere. From inside, it was whitewashed at the cost of ₹4989.60. If the cost of whitewashing is ₹20 per square meter, find the
(i) inside surface area of the dome,
(ii) volume of the air inside the dome

Answer 1

(i) Cost of whitewashing the dome from inside = Rs 498.96 
Cost of white washing 1m² area = Rs 2 
Therefore, CSA of the inner side of dome =\(\frac{4989.60}{20} \)= 249.48 m²

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(ii) Let the inner radius of the hemispherical dome be r. 
CSA of inner side of dome = 249.48 m² 
\(2\pi r^2\) = 249.48 m²

\(⇒\) r² = \(\frac{249.48}{2\pi}\) m²

\(⇒\) r² = 249.48 \(÷\) (2 ×\(\frac{ 22}{7}\))
\(⇒\) r² = 39.69
\(⇒\) r = \(\sqrt{39.69}\)
\(⇒\) r = 6.3 m
Volume of air inside the dome = Volume of hemispherical dome
\(=\frac{ 2}{3}\pi\)r³

\(= \frac{2}{3}\) × \(\frac{22}{7}\) × 6.3 m × 6.3 m × 6.3 m
= 523.9 m³ (approximately)
Therefore, the volume of air inside the dome is 523.9 m³ .