Question

The base radius of a cylinder is\[1\dfrac{2}{3}\] times its height. The cost of painting its curved surface area at \[2{\rm{ paisa/c}}{{\rm{m}}^2}\] is Rs92.40. find the volume of the cylinder.
A) \[80850{\rm{c}}{{\rm{m}}^3}\]
B) \[80580{\rm{c}}{{\rm{m}}^3}\]
C) \[80508{\rm{c}}{{\rm{m}}^3}\]
D) \[800508{\rm{c}}{{\rm{m}}^3}\]

Answer 1

Hint:
Here, we have to find the volume of the cylinder. Firstly we will find the area of the curved surface by dividing the total cost of painting i.e. Rs92.40 to the basic cost of painting i.e. \[2{\rm{ paisa/c}}{{\rm{m}}^2}\]. Then by equating the area of curved surface with the calculated value of area of curved surface, we will get the value of height and radius of the cylinder. Then by simply putting the value of radius and height of the cylinder in the formula of volume we will get the volume of the cylinder.

Complete step by step solution:
Let, r be the radius of the cylinder and h be the height of the cylinder.
It is given that the radius of the cylinder is equal to the \[1\dfrac{2}{3}\] times its height of the cylinder.
\[{\rm{r = 1}}\dfrac{2}{3}{\rm{h = }}\dfrac{5}{3}{\rm{h}}\]………. (1)
Now we have to find out the the area of the curved surface by dividing the total cost of painting i.e. Rs92.40 to the basic cost of painting i.e. \[2{\rm{ paisa/c}}{{\rm{m}}^2}\].
Therefore, area of the curved surface \[{\rm{ = }}\dfrac{{92.40}}{{0.02}} = 4620{\rm{ c}}{{\rm{m}}^2}\]
We know that area of the curved surface of cylinder\[{\rm{ = 2\pi rh = 4620}}\]
Now we have to solve the above equation to find out the height and radius of the cylinder.
\[ \Rightarrow {\rm{2}} \times {\rm{3}}{\rm{.14}} \times {\rm{r}} \times {\rm{h = 4620}}\]
We will put the value of r from the equation (1)
\[ \Rightarrow {\rm{2}} \times {\rm{3}}{\rm{.14}} \times \dfrac{5}{3}{\rm{h}} \times {\rm{h = 4620}}\]
\[ \Rightarrow {{\rm{h}}^2} = \dfrac{{{\rm{4620}} \times 3}}{{{\rm{2}} \times {\rm{3}}{\rm{.14}} \times 5}}\]
Now by solving this we will get the value of the height of the cylinder.
\[ \Rightarrow {{\rm{h}}^2}{\rm{ = 441}}\]
\[ \Rightarrow {\rm{h = 21 cm}}\]
Therefore height of the cylinder is equal to 21 cm.
Now by putting the value of h in equation (1) we will get the value of the radius of the cylinder.
\[ \Rightarrow {\rm{r = }}\dfrac{5}{3}{\rm{h = }}\dfrac{5}{3} \times 21 = 5 \times 7 = 35{\rm{ cm}}\]
Therefore radius of the cylinder is equal to 35 cm.
Now we know that Volume of the cylinder\[{\rm{ = \pi }}{{\rm{r}}^{\rm{2}}}{\rm{h}}\] where, r is the radius of the cylinder, h is the height of the cylinder
So, by putting the value of h and r in the above equation we will get the volume of the cylinder.
Therefore, Volume of the cylinder\[{\rm{ = \pi }}{{\rm{r}}^{\rm{2}}}{\rm{h = 3}}{\rm{.14}} \times {35^{\rm{2}}} \times {\rm{21 = 80850 c}}{{\rm{m}}^3}\]

Hence, \[{\rm{80850 c}}{{\rm{m}}^3}\]is the volume of the cylinder
So, option A is correct.

Note:
Volume is the amount of space occupied by an object in three-dimensional space. Volume is generally measured in cubic units.
Volume of sphere\[ = \dfrac{{{\rm{4\pi }}{{\rm{r}}^{\rm{3}}}}}{{\rm{3}}}\] where, r is the radius of the sphere
Volume of the cylinder\[{\rm{ = \pi }}{{\rm{r}}^{\rm{2}}}{\rm{h}}\] where, r is the radius of the cylinder, h is the height of the cylinder
It is important to write the units while solving a problem, it plays an important role.