Question

The slant height of a right circular cone is \[10\] m and its height is \[8\] m, then the area of its curved surface is
\[\left( A \right){\text{ }}80\pi {m^2}\]
\[\left( B \right){\text{ 6}}0\pi {m^2}\]
\[\left( C \right){\text{ 65}}\pi {m^2}\]
\[\left( D \right){\text{ 70}}\pi {m^2}\]

Answer 1

Hint: First try to remember the formula of the curved surface area of the cone. The formula is \[\pi rl\] . The value of slant height given to us and also the value of the height of the cone. So to find the curved surface area we will first find the value of the radius by using the formula \[l = \sqrt {{h^2} + {r^2}} \] .After this we will get the value of r . Then substitute the value of r and slant height in the area formula. By doing this we are able to find the curved surface area of the cone.

Complete step-by-step solution:
First let’s just see the diagram below:

Where r is the base radius of the cone , \[l\] is the slant height of the cone and h represents the height of the cone.
The formula to find the curved surface area of the cone \[ = \pi rl\]
It is given to us that the slant height of the right circular cone is \[10\] m and height of the cone is \[8\] m. We know that in general the formula to calculate the value of slant height is
\[l = \sqrt {{h^2} + {r^2}} \]
On squaring both sides we get
\[{l^2} = {h^2} + {r^2}\]
From here r square will be
\[ \Rightarrow {r^2} = {l^2} - {h^2}\]
Therefore, r becomes
\[ \Rightarrow r = \sqrt {{l^2} - {h^2}} \]
Put the given values of l and h here
\[ \Rightarrow r = \sqrt {{{\left( {10} \right)}^2} - {{\left( 8 \right)}^2}} \]
On opening the square terms we get
\[ \Rightarrow r = \sqrt {100 - 64} \]
On subtracting we get
\[ \Rightarrow r = \sqrt {36} \]
We know that thirty six is the perfect square of six. Therefore,
\[ \Rightarrow r = \pm 6\]
As r is the radius which cannot be negative as it is a measurement so \[ - 6\] will be rejected. Therefore the value of radius r is \[6\] m. Now
Curved Surface Area of cone \[ = {\text{ }}\pi rl\]
On substituting the values in the above formula we get
\[= {\text{ }}\pi \left( 6 \right)\left( {10} \right)\]
On multiplication it becomes
\[ = {\text{ 60}}\pi \]
That is, the curved surface area of the cone is \[{\text{60}}\pi {m^2}\] .
Hence, the correct option is \[\left( B \right){\text{ 6}}0\pi {m^2}\]

Note: Keep in mind that the formula to find the curved surface area of the cone is \[\pi rl\] . Also remember the formula \[l = \sqrt {{h^2} + {r^2}} \] . From this formula you can find the value of one variable when the other two variables are given just like we did in this question. Do the calculations carefully to avoid any silly mistake.