Question

I am unable to understand the solution of the following question . Can anyone help.. "A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid."

Answer 1

it says that assume a hemisphere ( half sphere) is cut out from a cube which have same value of edge length to that of diameter of hemisphere. you have to simply add the surface area of the hemisphere to the surface area of cube (a=2r).

Answer 2

We know that surface area of the remaining solid = Surface area of the cube - surface area of the base of hemisphere + Curved surface area of the hemisphere.  ---- (1)


Let the area of the cube be d.

Then the surface area of the cube = 6 * s^2

                                                          = 6 * d^2.

                                                          = 6d^2.   -------- (2)


Let the diameter of the hemisphere be d.

We know that Radius r = diameter/2 = d/2.

Then the curved surface area of the hemisphere = 2pir^2

                                                                                   = 2pi(d/2)^2 

                                                                                   = pid^2/2  ------- (3).


Then the Base area of the hemisphere = pir^2

                                                                  = pi(d/2)^2  ------ (4).


Substitute (1),(3),(4) in (1), we get

Surface area of the remaining solid = 6d^2 - pi(d^2/2) + pi(d^2/4)

                                                            = 6d^2 + pid^2/4

                                                            = d^2(6 + pi/4)
                                                           
                                                            = d^2(24 + pi/4)

                                                           = 1/4 d^2(pi + 24).


Hope this helps!