A cuboidal block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter that the hemisphere can have ? Also, find the surface area of the solid.

Cuboidal block of side 7 cm is surmounted by a hemisphere as shown in figure below:

Side of cuboidal block = 7 cm.

Greatest diameter of hemisphere = 7 cm.

Radius = $\dfrac{\text{Diameter}}{2}$

= $\dfrac{7}{2}$ = 3.5 cm.

Surface area of the hemisphere = $2πr^2$.

Putting values we get,

Surface area of the hemisphere = $2 \times \dfrac{22}{7} \times 3.5^2$

$= \dfrac{2 \times 22 \times 12.25}{7} \\[1em] = \dfrac{44 \times 12.25}{7} \\[1em] = \dfrac{539}{7} \\[1em] = 77 \text{ cm}^2.$

Surface area of the cube = 6a² = 6 x 7² = 6 × 49 = 294 cm².

Surface area of base of hemisphere = πr².

Putting values we get,

Surface area of base of hemisphere = $\dfrac{22}{7} \times (3.5)^2$

$= \dfrac{22}{7} \times 12.25 \\[1em] = 38.5 \text{ cm}^2.$

Surface area of solid = Surface area of cube + Surface area of hemisphere - Surface area of base of hemisphere = 294 + 77 - 38.5 = 332.5 cm².

Hence, the greatest diameter that the hemisphere can have is 7 cm and surface area of the solid is 332.5 cm².