Question

A circular cylinder is circumscribed about a right prism having a square base one meter on edge. The volume of the cylinder is r. Find the altitude of the cylinder.

Answer 1

Hint: See the circular cylinder from the top and try to recognize the right-angled triangle in the square base in order to evaluate the radius of the cylinder. Use the Pythagoras theorem to evaluate the radius of the circular cylinder.

Complete step-by-step solution:
Let the altitude of the circular cylinder is \[h\,m\].
Since the prism is of a square base, therefore, all the sides are equal.
We can see the top view of the cylinder in the following figure.

For a given circumscribed circle of centre O the \[\Delta OAB\] is the right-angled triangle.
Here, OA is the radius of the circle and let it be \[r\].
\[\angle ABO = {90^ \circ }\] and we know that the side opposite to the right angle is the hypotenuse. Therefore, AO is the hypotenuse and side of the square prism whose length is equal to $r$.
The perpendicular line from the centre of the circle divides the opposite chord into two equal parts.
Therefore, \[AB = 0.5\,m\]
OB is equal to the half of the side of the square base.
Therefore, $OB = 0.5\,m$
Now we use the Pythagoras theorem to determine the radius of a circular cylinder.
Pythagoras theorem states that the square of the hypotenuse is equal to the sum of the squares of two remaining sides which we called base and perpendicular.
Apply the Pythagoras theorem in \[\Delta OAB\].
$O{A^2} = O{B^2} + A{B^2}$
Substitute all the values.

$  {r^2} = {\left( {0.5} \right)^2} + {\left( {0.5} \right)^2} $

$  {r^2} = 2{\left( {0.5} \right)^2} $

$  {r^2} = 0.5 $

Since the volume is given in the question therefore, we apply the formula of volume of the circular cylinder to evaluate the altitude.
The volume of a circular cylinder with radius $r$ and height $h$ is \[V=\pi {r^2}h\].
$\Rightarrow h = \dfrac{V}{\pi {r^2}}$
Substituting the values of ${r^2}$ and evaluate the $h$. Use $\pi = 3.14$

\[  h = \dfrac{{6.283}}{{3.14 \times 0.5}} \]
\[ h \approx 4\,m  \]
Hence, the approximate length of the altitude right circular cylinder $4\,m$.

Note: In this type of questions use the circular base to evaluate the radius and be careful about using the Pythagoras theorem. Analyse the right-angled triangle properly. The value of $\pi = \dfrac{{22}}{7}$ can also be used.