Question

A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of $4989.60$. If the cost of white-washing is $20$ per square meter, find the inside surface area of the dome, the volume of the air inside the dome.

Answer 1

We know that Area white washed$\times$Cost of white wash$=$Total cost

Area white washed$\times 20=4989.60$

Area white washed $=\frac{4989.60}{20}=249.48‬‬m^2$

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

$\therefore ,$inner surface area of dome $=249.48‬‬m^2$

Volume of air inside dome$=$volume of hemisphere$=\frac{2}{3}\pi r^3$

Let the radius of dome be $r$m

Let us first find the radius using surface area

Surface area of dome$=249.48‬‬m^2$

$\Rightarrow 2\pi r^2=249.48‬‬m^2$

$\Rightarrow 2\times \frac{22}{7}\times r^2=249.48‬‬m^2$

$\Rightarrow r^2=\frac{249.48‬‬\times 7}{2\times 22}=39.69‬$

$\Rightarrow r=\sqrt{39.69‬}=6.3$m

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

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

$=\frac{2}{3}\times \frac{22}{7}\times 6.3‬\times 6.3\times 6.3m^3=523.908m^3$