
The ratio of volume of a cylinder: volume of cone: volume of hemisphere of same radius and same height is
a) $1:2:3$
b) $2:3:1$
c) $3:1:2$
d) $1:1:1$
Answer
483.6k+ views
Hint: Here first we assume radius and height for all the three shapes. Then put radius and height in the formula of volume of all the three shapes. Then find the ratio. Use formula for the volume of the cylinder,$V=\pi {{r}^{2}}h$ , Volume of the cone,$V=\frac{1}{3}\pi {{r}^{2}}h$ and Volume of the hemisphere,$V=\frac{2}{3}\pi {{r}^{2}}h$
Complete answer:
Let the radius of the base be r and height be h in cylinder, cone and hemisphere.
As we know that height of the hemisphere = radius of the hemisphere$\Rightarrow h=r$.
Volume of the cylinder = $\pi {{r}^{2}}h=\pi {{r}^{2}}r=\pi {{r}^{3}}$.
Volume of the cone = $\frac{1}{3}\pi {{r}^{2}}h=\frac{1}{3}\pi {{r}^{2}}r=\frac{1}{3}\pi {{r}^{3}}$ [taken r=h]
Volume of hemisphere = $\frac{2}{3}\pi {{r}^{3}}$.
So, now the ratio of the volumes of the cylinder, cone and hemisphere will be -
\[\Rightarrow \pi {{r}^{3}}:\frac{1}{3}\pi {{r}^{3}}:\frac{2}{3}\pi {{r}^{3}}\][Common factor $\pi {{r}^{3}}$ is removed from each terms]
\[\Rightarrow 1:\frac{1}{3}:\frac{2}{3}\][Multiplying all the terms with $''3''$ ]
\[=3:1:2\].
Therefore the required answer is - The ratio of volume of a cylinder: volume of cone: volume of hemisphere of same radius and same height is $3:1:2$
Hence, from the given multiple choices, the option C is the correct answer.
Note:: In such types of problems where every term is unknown take the help of the standard general formula, and accordingly follow the given conditions. Also, keep in mind which terms are taken in ratio as $\frac{1}{2}\text{ and }\frac{2}{1}$ make the major difference. So wisely take the ratio by converting the word statement in the proper mathematical form.
Complete answer:
Let the radius of the base be r and height be h in cylinder, cone and hemisphere.
As we know that height of the hemisphere = radius of the hemisphere$\Rightarrow h=r$.
Volume of the cylinder = $\pi {{r}^{2}}h=\pi {{r}^{2}}r=\pi {{r}^{3}}$.
Volume of the cone = $\frac{1}{3}\pi {{r}^{2}}h=\frac{1}{3}\pi {{r}^{2}}r=\frac{1}{3}\pi {{r}^{3}}$ [taken r=h]
Volume of hemisphere = $\frac{2}{3}\pi {{r}^{3}}$.
So, now the ratio of the volumes of the cylinder, cone and hemisphere will be -
\[\Rightarrow \pi {{r}^{3}}:\frac{1}{3}\pi {{r}^{3}}:\frac{2}{3}\pi {{r}^{3}}\][Common factor $\pi {{r}^{3}}$ is removed from each terms]
\[\Rightarrow 1:\frac{1}{3}:\frac{2}{3}\][Multiplying all the terms with $''3''$ ]
\[=3:1:2\].
Therefore the required answer is - The ratio of volume of a cylinder: volume of cone: volume of hemisphere of same radius and same height is $3:1:2$
Hence, from the given multiple choices, the option C is the correct answer.
Note:: In such types of problems where every term is unknown take the help of the standard general formula, and accordingly follow the given conditions. Also, keep in mind which terms are taken in ratio as $\frac{1}{2}\text{ and }\frac{2}{1}$ make the major difference. So wisely take the ratio by converting the word statement in the proper mathematical form.
Recently Updated Pages
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Explain the Treaty of Vienna of 1815 class 10 social science CBSE

Write an application to the principal requesting five class 10 english CBSE
