Question

The cost of painting the curved surface area of a cone at $5{\text{ ps/c}}{{\text{m}}^2}$is Rs. 35.20, find the volume of the cone if the slant height is 25cm.
A. $1223c{m^3}$
B. $1979c{m^3}$
C. $1323c{m^3}$
D. $1332c{m^3}$

Answer 1

Hint: Convert the given cost of painting from paise to rupees by dividing it by100. Find the curved surface of the cone by dividing the total cost to paint the cone by the cost of painting area of $1{\text{ c}}{{\text{m}}^2}$. Use the given slant height to find the radius of the cone. Then, calculate the volume of the cone using the formula.

Complete step by step answer:

First of all we will convert the cost of painting $1{\text{ c}}{{\text{m}}^2}$ from paise into rupees by dividing it by 100.
Hence, the cost of painting is $Rs.\dfrac{5}{{100}}{\text{ /c}}{{\text{m}}^2} = Rs.0.05{\text{per c}}{{\text{m}}^{\text{2}}}$
Now, we will find the curved surface of the cone by dividing the total cost to paint the cone by cost of painting area of $1{\text{ c}}{{\text{m}}^2}$.
Hence, the curved surface area of the cone is,
$\dfrac{{35.20}}{{0.05}} = 704c{m^2}$
Also, the curved surface area of the cone is given by, $A = \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height of the cone.
On substituting , $A = 704$, $l = 25$ and $\pi = \dfrac{{22}}{7}r$ to find the radius, $r$of the cone.
$
  704 = \dfrac{{22}}{7}r\left( {25} \right) \\
  r = \dfrac{{704\left( 7 \right)}}{{22\left( {25} \right)}} \\
  r = 8.96 \approx 9cm \\
$
Now, we will find the volume of the cone using the formula, $\dfrac{1}{3}\pi {r^2}h$
But, we need height of the cone to find the volume.
Height of cone can be calculated using the formula, $h = \sqrt {{l^2} - {r^2}} $
Thus, the height of the cone is
 $
  h = \sqrt {{{\left( {25} \right)}^2} - {{\left( 9 \right)}^2}} \\
  h = \sqrt {625 - 81} \\
  h = \sqrt {544} \\
  h = 23.32cm \\
 $
The volume of the cone is calculated using the formula, $\dfrac{1}{3}\pi {r^2}h$
$\dfrac{1}{3}\left( {\dfrac{{22}}{7}} \right){\left( 9 \right)^2}\left( {23.32} \right) = \dfrac{1}{3}\left( {\dfrac{{22}}{7}} \right)\left( {81} \right)\left( {23.32} \right) = 1979c{m^3}$
Hence, option B is correct.

Note: For these types of questions, one must remember the formulas of curved surface area and volume of the cone. Also, the height in the formula of the volume of the cone is the perpendicular height and not the slant height. Hence, in this question we have to first find the perpendicular height from the given slant height using the formula $h = \sqrt {{l^2} - {r^2}} $, where $h$ is the perpendicular height, $l$ is the slant height and $r$ is the radius of the cone.