Question

Find the capacity in litres of a conical vessel with
(i) Radius \[7cm\], slant height \[25cm\]
(ii) Height \[12cm\], slant height \[13cm\]

Answer 1

Hint:Use the formula for calculating the volume of the right circular cone which is \[\dfrac{1}{3}\pi {{r}^{2}}h\], where \[r\] is the radius of the cone and \[h\] is the height of the cone. When radius or height is not given, use the relation between radius, slant height (denoted by \[l\]), and height which is \[l=\sqrt{{{h}^{2}}+{{r}^{2}}}\] to find the radius or height of the cone and then calculate its volume.

Complete step-by-step answer:
We have to evaluate the volume of a right circular cone with given dimensions. We know that the volume of a cone whose height is \[h\] and radius is \[r\] is given by \[\dfrac{1}{3}\pi {{r}^{2}}h\].

(i) We have a cone whose radius is \[7cm\] and slant height is \[25cm\]. We will evaluate the height of the cone. We know the relation \[l=\sqrt{{{h}^{2}}+{{r}^{2}}}\] where \[l\] is the slant height of the cone, \[r\] is the radius of the cone and \[h\] is the height of the cone.
Substituting \[r=7cm,l=25cm\] in the above equation, we have \[25=\sqrt{{{h}^{2}}+{{\left( 7 \right)}^{2}}}\]. Simplifying the equation, we have \[{{\left( 25 \right)}^{2}}={{h}^{2}}+{{\left( 7 \right)}^{2}}\].
\[\begin{align}
  & \Rightarrow {{h}^{2}}=625-49=576 \\
 & \Rightarrow h=24cm \\
\end{align}\]
Now, we will evaluate the volume of the cone.
Substituting the value \[r=7cm,h=24cm\] in the formula \[\dfrac{1}{3}\pi {{r}^{2}}h\], we have the volume of the cone \[=\dfrac{1}{3}\times \dfrac{22}{7}{{\left( 7 \right)}^{2}}\times 24=22\times 7\times 8=1232c{{m}^{3}}\].
Hence, the volume of the cone whose dimensions \[r=7cm,l=25cm\] is \[1232c{{m}^{3}}\].
(ii) ) We have a cone whose height is \[12cm\] and slant height is \[13cm\]. We will evaluate the radius of the cone. We know the relation \[l=\sqrt{{{h}^{2}}+{{r}^{2}}}\] where \[l\] is the slant height of the cone, \[r\] is the radius of the cone and \[h\] is the height of the cone.
Substituting \[h=12cm,l=13cm\] in the above equation, we have \[13=\sqrt{{{r}^{2}}+{{\left( 12 \right)}^{2}}}\]. Simplifying the equation, we have \[{{\left( 13 \right)}^{2}}={{r}^{2}}+{{\left( 12 \right)}^{2}}\].
\[\begin{align}
  & \Rightarrow {{r}^{2}}=169-144=25 \\
 & \Rightarrow r=5cm \\
\end{align}\]
Now, we will evaluate the volume of the cone.
Substituting the value \[r=5cm,h=12cm\] in the formula \[\dfrac{1}{3}\pi {{r}^{2}}h\], we have the volume of the cone \[=\dfrac{1}{3}\times \dfrac{22}{7}{{\left( 5 \right)}^{2}}\times 12=314.28c{{m}^{3}}\].
Hence, the volume of the cone with dimensions \[h=12cm,l=13cm\] is \[314.28c{{m}^{3}}\].

Note: Be careful about the units while calculating the volume of the cone; otherwise we will get an incorrect answer. A right circular cone is a cone where the axis of the cone is the line meeting the vertex to the midpoint of the circular base.