Question

The radius and height of a cone are each increased by 20 % then the volume of the cone is increased by
A. 20%
B. 40%
C. 60%
D. 72.8%

Answer 1

Hint: First assume the base radius and height of cone as r and h respectively.
Then, calculate the increased base radius and increased height of the cone by using percentage concept in the original radius and height of the cone. After that, substitute the value of the increased base radius and increased height of the cone in the original volume of the cone.
The volume of cone is given by:
\[V=\ \dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\]
Finally, using the increase percentage formula calculate the increased percentage volume of the cone.

Complete step-by-step answer:
According to the question, it has been given that:
The radius of a cone is increased by 20% and
The height of a cone is increased by 20%.
The formula of the volume of a cone with base radius `r' and height ` h' is given by:
\[V=\ \dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h----(1)}\]
Let equation1 be the original Volume and let it be denoted as `V'.
As, it is given that the base radius and height of the cone is increased by 20%. So, now let's calculate the value of the increased base radius and increased height of the cone.
If base radius of the original cone `r' is increased 20%, then new radius~will be calculated as:
$\Rightarrow$ \[new\ radius=\ r\ \times \ \dfrac{\left(100\ +increased\%\ \right)}{100}\]
\[\Rightarrow new\ radius=\ r\ \times \ \dfrac{\left(100\ +20\ \right)}{100}\]
\[\Rightarrow new\ radius=\ r\ \times \ \dfrac{120}{100}\ \Rightarrow new\ radius=\ \dfrac{6}{5}r\ cm\ ---\left(2\right)\]
If the height of the original cone ` h' is increased 20%, then the new height~will be calculated as:
\[new\ height=h\ \times \ \dfrac{\left(100\ +increased\%\ \right)}{100}\]
\[\Rightarrow new\ height=h\ \times \ \dfrac{\left(100\ +20\ \right)}{100}\]
\[\Rightarrow new\ height=h\ \times \ \dfrac{120}{100}\ \Rightarrow new\ height=\ \dfrac{6}{5}h\ cm\ ---\left(3\right)\]
Now, in order to calculate the volume of the new cone (V'),

Substitute equation 2 and 3 in equation 1 and we get:
$\Rightarrow$\[\dfrac{1}{3}\mathrm{\pi}{\left(\dfrac{6}{5}\mathrm{r}\right)}^2\left(\dfrac{6}{5}\mathrm{h}\right)\ =\dfrac{216}{125}\left(\dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\right)\ =\ V'----(4)\]
Now, the increased percentage volume of the cone can be calculated as:
\[therefore % increased volume= \dfrac{V'-V}{V}\times 100\ ----\left(5\right)\]
Substitute equation 1 and 4 in equation 5
\[\therefore % increased volume= \dfrac{\dfrac{216}{125}\left(\dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\right)-\left(\dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\right)}{\left(\dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\right)}\times 100\ \]
\[\Rightarrow \%\ increased\ volume=\ \dfrac{\left(\dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\right)\left(\dfrac{216}{125}-1\right)}{\left(\dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\right)}\times 100\]
\[$\Rightarrow$ \%\ increased\ volume=\ \dfrac{\left(\dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\right)\left(\dfrac{91}{125}\right)}{\left(\dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\right)}\times 100\ =\ \dfrac{91}{125}\times 100\]
\[$\Rightarrow$ \%\ increased\ volume=\ \boldsymbol{\mathrm{72}}.\boldsymbol{\mathrm{8}}\boldsymbol{\mathrm{\%}}\]

Therefore, the correct option for the given question is D

Note:
The formula of volume of a cone with base radius `r' and height ` h' is given by:
$\Rightarrow$\[V=\dfrac{1}{3}{\mathrm{\pi }\mathrm{r}}^2\mathrm{h}\]
Thus, remembering the above formula, we can determine the volume of a cone for any given question.
Also, for the percentage increment can be calculated by using the formula:
\[\therefore % increased= \dfrac{\mathrm{modified\ form}-original\ form}{original\ form}\times 100\]