Question

The cost of painting the total surface area of the cone at 25 paise per cm square is Rs. 176. Find the volume of the cone, if its slant height is 25 cm.

Answer 1

Hint: We will use the formula to evaluate the total surface area of the cone by dividing the total cost of the painting by the rate of charge. We are having the slant height and has to calculate the radius, so, we will use the total surface area of the cone formula, \[TSA = \pi r\left( {l + r} \right)\]. After finding the value of the radius, we will find the height of the cone by applying the Pythagoras theorem and in last apply the formula of volume of cone and substitute the values in it.

Complete step by step solution:
First, consider the given total cost of painting the cone which is equal to Rs. 176
As we are given the rate of charge as 25 paise per cm square, we will find the total surface area of the cone by dividing the total cost of painting cone by the rate of charge,
Thus, we get,
\[ \Rightarrow {\text{Total surface area of cone}} = \dfrac{{{\text{total cost of painitingcone}}}}{{{\text{rate of charge}}}}\]
Now, we will substitute the values in the formula above to determine the total surface area,
Hence,
\[ \Rightarrow \dfrac{{176}}{{0.25}} = 704\]
Thus, we get the total surface area of the cone as \[704\]cm square.
Next, let the radius of the cone be \[r\] cm and the slant height of the cone be 25 cm which is denoted by \[l\].
Since we know the formula for the total surface area of the cone as \[\pi r\left( {l + r} \right)\] and put it equal to 704
Thus, we get,
\[ \Rightarrow \pi r\left( {l + r} \right) = 704\]
Next, we will substitute the values of \[\pi \] and \[l\] to evaluate the values of \[r\].
\[
   \Rightarrow \left( {\dfrac{{22}}{7}} \right)r\left( {25 + r} \right) = 704 \\
   \Rightarrow {r^2} + 25r = \dfrac{{704\left( 7 \right)}}{{22}} \\
   \Rightarrow {r^2} + 25r = 224 \\
 \]
Now, we will use the middle term splitting method to simplify it further,
Thus, we get,
\[
   \Rightarrow {r^2} + 32r - 7r - 224 = 0 \\
   \Rightarrow r\left( {r + 32} \right) - 7\left( {r + 32} \right) = 0 \\
   \Rightarrow \left( {r + 32} \right)\left( {r - 7} \right) = 0 \\
 \]
Thus, apply the zero-factor property and find the values of \[r\]
\[ \Rightarrow r + 32 = 0\] and \[r - 7 = 0\]
\[ \Rightarrow r = - 32\] and \[r = 7\]
Here, we will ignore the negative value and consider the positive value.
Therefore, the radius of the cone is \[r = 7\] cm.
Next, we will use the Pythagoras theorem to evaluate the value of the height of the cone.
In this, height represents the perpendicular, length represents the hypotenuse and radius represents the base.
Thus, we get,
\[h = \sqrt {{l^2} - {r^2}} \]
Substituting the values, we get,
\[
   \Rightarrow h = \sqrt {{{\left( {25} \right)}^2} - {{\left( 7 \right)}^2}} \\
   \Rightarrow h = \sqrt {625 - 49} \\
   \Rightarrow h = \sqrt {576} \\
   \Rightarrow h = 24 \\
 \]
Thus, the height of the cone is 24 cm.
Now, we will apply the formula of the volume of a cone given by \[V = \dfrac{1}{3}\pi {r^2}h\] and substitute the values,
Thus, we get,
\[
  V = \dfrac{1}{3}\left( {\dfrac{{22}}{7}} \right)\left( {{7^2}} \right)\left( {24} \right) \\
   = 22\left( 7 \right)\left( 8 \right) \\
   = 1232 \\
 \]
Hence, the volume of the cone is given by a 1232 cm cube.

Note: The rate of charge is given as 25 paise per square cm that is why we have used rate of charge equals to \[0.25\]. We have ignored the negative value of radius as radius can never be negative. We can calculate the total cost also if it is not given and total surface area and rate of charge are given. Remember the formulas for the total surface area of the cone and volume of the cone.