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Question

A dome of a building is in the form of a hemisphere. From inside, it was white-washed, at the cost of Rs. $$498.96$$. If the cost of white-washing is $$Rs. 2.00$$ per square metre, find the 
(i) inside surface area of the dome
(ii) volume of the air inside the dome.


Solution

(i) We know that

Area whitewashed $$\times$$ Cost of white wash $$=$$ Total cost

Area whitewashed $$\times2=498.96$$

Area whitewashed $$=\dfrac{498.96}{2}$$

Area whitewashed $$=249.48\ m^2$$

Now,

Area of whitewashed is curved surface area of hemisphere (as only walls are whitewashed, not floor)

Therefore,

Inner surface area of dome $$=249.48\ m^2$$

(ii) Volume of air inside dome $$=$$ Volume of hemisphere

                                                  $$=\dfrac{2}{3}\pi r^3$$

Let the radius of dome be 'r' m
Surface area of dome $$=249.48\ m^2$$

                              $$2\pi r^2=249.48$$

              $$2\times\dfrac{22}{7}\times r^2=249.48$$

                                $$r^2=39.69$$

                                $$r=\sqrt{39.69}$$

                                $$r=6.3\ m$$

Volume of air inside the dome $$=$$ Volume of hemisphere

                                                    $$=\dfrac{2}{3}\pi r^3$$

                                                    $$=\dfrac{2}{3}\times\dfrac{22}{7}\times6.3^3$$

                                                    $$=523.9\ m^3$$

Mathematics

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