A solid cube has sides of length 4 cm. A hemisphere of radius 1.5 cm is removed from the cube as shown in the figure. Calculate the total surface area, in cm^2 to 3 significant figures, of the remaining solid.
I got the total surface area of the cube = 6(a)^2 = 96 cm^2 The total surface area of a hemisphere = 3pi(r)^2 = 21.2 cm^2 If I subtract them both to find the tsa of the remaining solid = 96 - 21.2 = 74.8 cm^2 but THE ANSWER IS APPARENTLY 103 cm^2. HOW??
We are talking about surface area and not volume. It is a solid cube and if you cut out a hemisphere, the surface area of resulting shape in fact increases, instead of decreasing.
Net surface area ($S$) = Surface Area of cube ($S_1$) - area of circle in the plane where hemisphere is cut out ($S_2$) + surface area of hemisphere ($S_3)$.
If $a$ is the side of the cube and $r$ is the radius of the hemisphere cut out (where $r \leq \frac{a}{2}$ if the center of the hemisphere is at the center of one of the faces of the cube),
$S = 6 a^2 - \pi r^2 + 2\pi r^2 = 6a^2 + \pi r^2$