Question

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter $d$ of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining block.

• A. $\frac{1}{4}{d}^2(\pi +16)$
• B. $\frac{1}{4}{d}^2(\pi +24)$
  
• C. $\frac{1}{8}{d}^2(\pi +16)$
• D. $\frac{1}{8}{d}^2(\pi +24)$

Answer 1

Answer: (B)

$\frac{1}{4}{d}^2(\pi +24)$

After cutting out a hemispherical depression, the view of the surface looks like this. This diameter of the hemisphere is d. Thus, the side of the cube is also d. The surface area of each of the other $5$ faces of the cube is $5d^2$.

For the face from which a hemisphere has been cur, subtract the area of a circle if diameter d from $d^2$ and then add the area of a hemisphere of diameter d to it.

Area of circle of diameter d = $\pi \frac{d^2}{4}$

Surface area of a hemisphere of diameter d = $2\pi \frac{d^2}{4}$

That is the surface area of the face from which a hemisphere has been cut is:

$d^2-\frac{\pi d^2}{4}+2\frac{\pi d^2}{4}=d^2+\pi \frac{d^2}{4}$

Total surface area of the remaining solid$=5d^2+d^2+\frac{\pi d^2}{4}$

$=6d^2+\frac{\pi d^2}{4}$

$=\frac{1}{4}{d}^2(\pi +24)$