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 (i) inside surface area of the dome (ii) volume of the air inside the dome (Assume $\mathrm{\pi }=\frac{22}{7}$)

Answer 1

Step 1: Finding the curved surface area

Cost of white-washing the dome from inside$=₹4989.60$

The Cost of white-washing area per square meter$=₹20$

Therefore,

$\begin{array}{rcl}\mathrm{Curved}\mathrm{surface}\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{inner}\mathrm{side}\mathrm{of}\mathrm{dome}& =& \frac{\mathrm{Total}\mathrm{cost}}{\mathrm{Cost}\mathrm{per}\mathrm{meter}\mathrm{square}}\\ & =& \frac{4989.60}{20}\\ & =& 249.48{\mathrm{m}}^2\end{array}$

Step 2: Finding the radius of the dome

Let us assume the inner radius of the hemispherical dome be $\mathrm{r}$.

Curved surface area of the inner side of dome$=249.48{\mathrm{m}}^2$

We know that,

$$\begin{gathered} \mathrm{CSA}\mathrm{of}\mathrm{a}\mathrm{hemisphere}=2{\mathrm{\pi r}}^2 \\ \Rightarrow 2{\mathrm{\pi r}}^2=249.48 \\ \Rightarrow 2\times \frac{22}{7}\times {\mathrm{r}}^2=249.48 \\ \Rightarrow {\mathrm{r}}^2=\frac{249.48\times 7}{2\times 22} \\ \Rightarrow {\mathrm{r}}^2=39.69 \\ \Rightarrow \mathrm{r}=6.3 \end{gathered}$$

Hence, the radius $\mathrm{r}=6.3\mathrm{m}$

Step 3: Finding the volume of the dome

The volume of air inside the dome = Volume of hemispherical dome

We know that,

$\begin{array}{rcl}\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{hemisphere}& =& \frac{2}{3}{\mathrm{\pi r}}^3\\ & =& \frac{2}{3}\times \frac{22}{7}\times {(6.3)}^3=523.908{\mathrm{m}}^3\\ & =& 523.9{\mathrm{m}}^3(\mathrm{approx}.)\end{array}$

Therefore, the volume of air inside the dome is $523.9{\mathrm{m}}^3$