Question

A dome of a building is in the form of a hemisphere. From inside, it was whitewashed at the cost of Rs 498.96. if the cost of whitewashing is Rs 2.00 per square meter, find
i. the inside surface area of the dome
ii. volume of the air inside the dome.

Answer 1

Hint: In such a question, the total cost of whitewashing is given to us and the cost of white washing per square meter is also given. Simply calculate the surface area whitewashed which is curved surface area. Use formula for curved surface area of Hemisphere to find the radius and calculate volume using direct formula for the volume of the hemisphere.

Complete step-by-step answer:
i. Cost of white washing of dome(C) = Rs. 498.96

Cost of white washing per square meter(p) = Rs. 2

Now, Curved surface area of the hemispherical dome $ = \dfrac{C}{p}$

$ = \dfrac{{Rs.498.96}}{{\left( {\dfrac{{Rs.2}}{{{m^2}}}} \right)}}$

$ = 249.48{m^2}$

Hence, the Curved surface area of the hemispherical dome or the inside surface area of the dome $ = 249.48{m^2}$

ii. Let us suppose ‘r’ be the radius of the Hemispherical dome.

We know that Curved surface area of the Hemisphere $ = 2\pi {r^2}$

So, we can say that

$ \Rightarrow 2\pi {r^2} = 249.48{m^2}$

Divide both sides by $2\pi $ we get,

$ \Rightarrow {r^2} = \dfrac{{249.48{m^2}}}{{2\pi }}$

On substituting $\pi = \dfrac{{22}}{7}$ and simplifying we get,

$ \Rightarrow {r^2} = 39.7{m^2}$

Taking Square root both sides we get,

$ \Rightarrow r = 6.3m$

Now, the radius ‘r’ of the hemispherical dome is = 6.3m

We know that, Volume of the hemisphere $ = \dfrac{2}{3}\pi {r^3}$

Now, Volume of Air inside the dome = Volume of the Hemispherical dome $ = \dfrac{2}{3}\pi {r^3} = \dfrac{2}{3} \times \dfrac{{22}}{7} \times {\left( {6.3m} \right)^3} = 523.9{m^3}$

Hence, the volume of air inside the dome $ = 523.9{m^3}$

Note- For such types of questions just evaluate the question according to the approach by using formulas of mensuration like Curved surface area of the Hemisphere is $ = 2\pi {r^2}$ and the Volume of the Hemisphere is $ = \dfrac{2}{3}\pi {r^3}$ . Then just simply find the radius according to one equation of curved surface area and use that to calculate the parameters like volume.