Question

A cubical block of side $7cm$ is surmounted by a hemisphere of the largest size. Find the surface area of the resultant solid.

Answer 1

Consider the problem

The hemisphere can occupy whole of the side of cube.

Therefore, Greatest diameter of hemisphere $=sideofcube$ $=7cm$

Here, base of hemisphere falls on cube, so that area should not from part of solid.

$Surfaceareaofsolid=Areaofcube+curvedsurfaceareaofhemisphere-baseareaofhemisphere$

Area of cube $=6a^2$

Here, $a=side=7cm$

So,

$\begin{array}{c}=6{\left(7\right)}^2\\ =6\times \left(7\times 7\right)\\ =6\times 49\\ =294c{m}^2\end{array}$

And,

Curved surface area of hemisphere $=2\pi r^2$

Diameter of hemisphere$=7cm$

Therefore, radius is

$\begin{array}{c}r=\frac{diameter}{2}\\ =\frac{7}{2}cm\end{array}$

So,

$\begin{array}{c}=2\times \frac{22}{7}\times {\left(\frac{7}{2}\right)}^2\\ =2\times \frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\\ =77c{m}^2\end{array}$

And,

Base area of hemisphere $=\pi r^2$

$\begin{array}{c}=\frac{22}{7}\times {\left(\frac{7}{2}\right)}^2\\ =\frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\\ =\frac{77}{2}c{m}^2\end{array}$

Now,

$Surfaceareaofsolid=Areaofcube+curvedsurfaceareaofhemisphere-baseareaofhemisphere$

$\begin{array}{c}=294+77-\frac{77}{2}\\ =294+77-38.5\\ =371-38.5\\ =332.5c{m}^2\end{array}$

Hence, surface area of solid $=332.5c{m}^2$