Question

If the volume of the cylinder is equal to \[12436c{m^3}\] and radius and height are in the ratio 2:3 respectively. Find its height.
A. 21cm
B. 7cm
C. 14cm
D. 18cm

Answer 1

Hint: Here we are given with the volume of the cylinder directly. Also we are given the ratio of radius to height of the cylinder. We will rearrange this ratio and we will convert radius in the form of height. And then putting these values in formula we will find the height of the cylinder.

Formula used:
Volume of cylinder = \[\pi {r^2}h\]

Step by step solution:
Given that r:h=2:3
\[ \Rightarrow r = \dfrac{2}{3}h\]
And volume of cylinder is \[12436c{m^3}\]
Thus using the formula now,
Volume of cylinder = \[\pi {r^2}h\]
\[ \Rightarrow 12436 = \dfrac{{22}}{7} \times {\left( {\dfrac{2}{3}h} \right)^2} \times h\]
Taking the square,
\[ \Rightarrow 12436 = \dfrac{{22}}{7} \times \dfrac{4}{9}{h^2} \times h\]
Rearranging the terms and in order to find the height,
\[ \Rightarrow {h^3} = 12436 \times \dfrac{9}{4} \times \dfrac{7}{{22}}\]
On multiplying the terms we get,
\[ \Rightarrow {h^3} = \dfrac{{12436 \times 63}}{{88}}\]
\[ \Rightarrow {h^3} = \dfrac{{783468}}{{88}}\]
On dividing the ratio,
\[ \Rightarrow {h^3} = 8903.045\]
Taking the cube root on both sides,
\[ \Rightarrow h = \sqrt[3]{{8903.045}}\]
\[ \Rightarrow h = 20.72\]
But on observing the options we will round off the answer to
\[ \Rightarrow h \approx 21cm\]
This is the height of the cylinder \[ \Rightarrow h \approx 21cm\]

Thus option A is the correct answer.

Note:
Note that we converted radius in the form of height because we have to find the height in the question. If we were asked to find the radius then we would have reversed the arrangement. Also the calculation method or pattern may differ but the answer should be the same.