Question

From a solid cylinder of height 4 cm and radius 3 cm, a conical cavity of height 4 cm and of base radius 3 cm is hollowed out. What is the total surface area of the remaining solid?
A. \[15\pi \] square cm
B. \[22\pi \] square cm
C. \[33\pi \] square cm
D. \[48\pi \] square cm

Answer 1

Hint: First of all, find the slant height of the conical cavity. Then the remaining total surface area of the remaining solid is the sum of the curved surface areas of the conical cavity and cylinder and the area of the base of the cylinder. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:

Given,

Radius of the cylinder \[r = 3{\text{ cm}}\]

Radius of conical cavity \[r = 3{\text{ cm}}\]

Height of the cylinder \[h = 4{\text{ cm}}\]

Height of the conical cavity \[h = 4{\text{ cm}}\]

We know that the slant height of the conical cavity of radius \[r\] and height \[h\] is given by \[l = \sqrt {{r^2} + {h^2}} \] as shown in the below figure:

Let \[l\] be the slant of the conical cavity which is given by

\[l = \sqrt {{3^2} + {4^2}} = \sqrt {9 + 16} = \sqrt {25} = 5{\text{ cm}}\]

We know that the curved surface area of the conical cavity of radius \[r\] and slant height
\[l\] is given by\[\pi rl\]. The curved surface area of the cylinder of radius \[r\] and height
\[h\] is given by \[2\pi rh\]. And the area of the base of the cylinder of radius \[r\] is given by
\[\pi {r^2}\].

From the figure, it is clear that

Total surface area of remaining solid = curved surface area of conical cavity + curved surface area of cylinder + area of base of cylinder.

Therefore, total surface area of remaining solid is

\[

   = \pi rl + 2\pi rh + \pi {r^2} \\

   = \pi r\left( {l + 2h + r} \right) \\

   = \pi 3\left( {5 + 2 \times 4 + 3} \right) \\

   = 3\pi \left( {16} \right) \\

   = 48\pi \\

\]

Hence the total surface area of remaining solid is \[48\pi \] square cm

Thus, the correct option is D. \[48\pi \] square cm

Note: The curved surface area of the conical cavity of radius \[r\] and slant height \[l\] is given by\[\pi rl\]. The curved surface area of the cylinder of radius \[r\] and height \[h\] is given by \[2\pi rh\]. And the area of the base of the cylinder of radius \[r\] is given by \[\pi {r^2}\].