Question

A hemispherical depression is cut out from one face of a cubical wooden block of edge 21 cm, such that the hemisphere is equal to the edge of the cube.Determine the volume and total surface area of the remaining block.

Answer 1

Solution:

As per the data given in the question itself, we have

Edge of the cubical wooden block = e = 21 cm

Diameter of the hemisphere = Edge of the cubical wooden block = 21 cm

Radius of the hemisphere = 10.5 cm = r (as we know that the radius is half of the diameter)

Now,

Volume of the remaining block = Volume of the cubical block – Volume of the hemisphere

V = $e^3-\left(\frac{2}{3}\pi r^3\right)={21}^3-(\frac{2}{3}\pi \times {10.5}^3)=6835.5c{m}^3$

Surface area of the block = SA = $6a^2=6\times {21}^2$ ………………… (1)

Curved surface area of the hemisphere = CSA = $2\pi r^2=2\pi \times {10.5}^2$ ……………………. (2)

Base area of the hemisphere = BA = $\pi r^2=\pi \times {10.5}^2$ …………………….(3)

So, Remaining surface area of the box = SA – (CSA + BA) = $6\times {21}^2-(2\pi \times {10.5}^2+\pi \times {10.5}^2)=2992.5c{m}^2$

Therefore, the remaining surface area of the block = 2992.5 cm²

Volume of the remaining block = V = 6835.5 cm³