4. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

It is given that hemisphere is mounted on the cuboid, thus the hemisphere can take on complete as its diameter (which is maximum).

Thus the greatest diameter of the hemisphere is 7 cm.

Now, the total surface area of solid  =   Surface area of cube  +   Surface area of the hemisphere -  Area of the base of a hemisphere (as this is counted on one side of the cube)

The surface area of the cube is :

$=\ 6a^3$

$=\ 6\times 7^3\ =\ 294\ cm^2$

Now the area of a hemisphere is

$=\ 2\pi r^2$

$=\ 2\times \frac{22}{7}\times \left ( \frac{7}{2} \right )^2\ =\ 77\ cm^2$

And the area of the base of a hemisphere is

$=\ \pi r^2\ =\ \frac{22}{7}\times \left ( \frac{7}{2} \right )^2\ =\ 38.5\ cm^2$

Hence the surface area of solid is   $= 294 + 77 - 38.5 = 332.5 \:cm^2$.

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