Question

From a solid cylinder whose height is $8cm$ and radius $6cm$, a conical cavity of height $8cm$ and of base radius $6cm$ is hollowed out. Find the volume of the remaining solid. Also, find the total surface area of the remaining solid. Take $\pi = 3.14$

Answer 1

Hint: First, we need to know about the concept of the volume of the cylinder and the volume of the cone to calculate the given question using the formula. Also, we have to calculate the volume of the remaining solid is exactly equal to the difference of the volume of the solid cylinder and the volume of the two conical holes.
Formula to be used:
The volume of the cylinder $ = \pi {r^2}h$
The volume of the cone $ = \dfrac{{\pi {r^2}h}}{3}$
where $h$ is the height of the given cylinder and $r$ is the radius

Complete step by step answer:
Since the solid cylinder height is $8cm$ and also given the radius is $6cm$. For the conical cavity, the height is given as $8cm$ and radius is given as $6cm$

Hence using the volume of the cylinder formula, we have $\pi {r^2}h$ where $h$ is the height of the given cylinder and $r$ is the radius
Substituting the know values solid cylinder height is $8cm$ and the radius is $6cm$
Hence, we get $\pi {r^2}h = \pi {(6)^2}(8)$ also the value of $\pi = 3.14$
Thus, we have $\pi {r^2}h = (3.14){(6)^2}(8)$
Further solving using multiplication we get $\pi {r^2}h = (3.14)(36)(8) \Rightarrow 904.32$ which is the value of the solid cylinder.
Now to find the volume of the conical cavity we have the formula as $\dfrac{{\pi {r^2}h}}{3}$
Thus, applying the values, we have $\dfrac{{\pi {r^2}h}}{3} = \dfrac{{(3.14){{(6)}^2}(8)}}{3}$
Further solving we have $\dfrac{{(3.14){{(6)}^2}(8)}}{3} = \dfrac{{904.32}}{3} \Rightarrow 301.44$ which is the value of the conical cavity
Hence to find the Volume of the resulting solid, we subtract the volume of the cylinder into the volume of the conical cavity.
Hence, we get $904.32 - 301.44 = 602.88$ which is the remaining solid of the surface area.

Note:
Both the formulas of the volume of the cylinder and volume of the conical cavity has the only difference, that is $\dfrac{1}{3}$ because of the volume of the cylinder formula we have $\pi {r^2}h$ and the volume of the conical cavity we have the formula as $\dfrac{{\pi {r^2}h}}{3}$
Thus, to find the total area of the surface we subtract the two terms.