A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of Rs. 4989.60. If the cost of white-washing is Rs 20 per square meter, find the
(i) inside surface area of the dome
(ii) volume of the air inside the dome
(Assume π = 22/7)
(i) Cost of white-washing the dome from inside = Rs 4989.60
Cost of white-washing 1 m2 area = Rs 20
CSA of the inner side of dome = \(\frac{498.96}{2}\) m2
= 249.48 m2
(ii) Let the inner radius of the hemispherical dome be r.
CSA of inner side of dome = 249.48 m2 (from (i))
Formula to find CSA of a hemi sphere = 2πr2
2πr2 = 249.48
2 × \(\frac{22}{7}\) × r2 = 249.48
r2 = \(\frac{249.48 \times 7}{2 \times 22}\)
r2 = 39.69
r = 6.3
So, radius is 6.3 m
Volume of air inside the dome = Volume of hemispherical dome
Using formula, volume of the hemisphere = \(\frac{2}{3}\) πr3
= (\(\frac{2}{3}\) × \(\frac{22}{7}\) × 6.3 × 6.3 × 6.3
= 523.908
= 523.9 (approx.)
Volume of air inside the dome is 523.9 m3.