Question

A hemisphere is cut out from one face of a cubical wooden block such that the diameter of the hemisphere is equal to length of the cube. Determine the surface area of remaining solid

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Answer 1

Let Initial surface area of the cube be a

a = 6 × length × length

Let Surface area of the hemisphere be b

b = 3 ×22/7 × radius × radius

Let Final surface area of the cube be c

c = a-b

Since diameter = length

2×Radius = length

In terms of radius of the hemisphere

c = 6×2r×2r - (3×22/7 ×r×r)

c = 6×r×r(4-11/7)

c = 6×r×r×17/7

c = r×r×102/7

In terms of length of the cube

c = l/2 × l\2 × 102/7

c = l×l×51/14

Please forgive me if It's incorrect

:-)

Answer 2

Diameter of the hemispherical depression = Edge of the cube = l

Radius of the hemispherical depression = $\sf{\frac{l}{2}}$

Total surface area of the solid after the hemispherical depression is cut

= TSA of the cube + CSA of the hemisphere - Top area of the hemisphere

= $\sf\red{{6l}^{2} + 2\pi {r}^{2} - \pi {r}^{2} = {6l}^{2} + \pi {r}^{2} }$

= $\sf\red{ {6l}^{2} + \pi {( \frac{l}{2} )}^{2}}$

= $\sf\red{\frac{ {l}^{2} }{4} (24 + \pi)square \: units}$