Question

If the volume of a right circular cylinder with its height equal to the radius is \[25\dfrac{1}{7}c{{m}^{3}}\], then the radius of the cylinder is equal to
(a) \[\pi \]
(b) 3 cm
(c) 4 m
(d) 2 cm

Answer 1

Hint: A cylinder with height h and radius r looks like this:

The volume of the cylinder is given by the formula \[V=\pi {{r}^{2}}h\], where r is the radius of the cylinder and h is the height. As we know that the diameter = 2 \[\times \] radius. Therefore, the volume of the cylinder can also be written as,
\[V=\pi {{\left( \dfrac{d}{2} \right)}^{2}}h\]
\[V=\dfrac{\pi }{4}{{d}^{2}}h\]

Complete step-by-step answer:
According to the question, we have a right circular cylinder with the height equal to the radius of the cylinder. So, the required diagram is given as shown below.

From here, h = r.
The formula of the volume of the cylinder denoted by ‘V’ results in the form \[V=\pi {{r}^{2}}h\]. Since the height is equal to the radius, this means h = r. Thus, we have a new expression for the volume of the cylinder which is given by \[V=\pi {{h}^{2}}h\]. Since it is given in the question itself that the volume of the cylinder is given by \[25\dfrac{1}{7}c{{m}^{3}}\]. Therefore, we get a new form of the volume of the cylinder as
\[\pi {{h}^{2}}h=25\dfrac{1}{7}\]
Now, we will multiply \[{{h}^{2}}\] and h together. Therefore, we have
\[\pi {{h}^{3}}=25\dfrac{1}{7}.....\left( i \right)\]
We can either substitute the value of \[\pi =\dfrac{22}{7}\] or \[\pi =3.14\]. Here, we are putting \[\pi =\dfrac{22}{7}\] in the equation (i), thus we get,
\[\dfrac{22}{7}{{h}^{3}}=25\dfrac{1}{7}\]
Now, we will multiply both the sides by \[\dfrac{7}{22}\] and we get a new expression written as
\[\dfrac{7}{22}\times \dfrac{22}{7}{{h}^{3}}=\dfrac{7}{22}\times 25\dfrac{1}{7}\]
This implies to a new equation after canceling the terms which is given by
\[{{h}^{3}}=\dfrac{7}{22}\times 25\dfrac{1}{7}\]
Since \[25\dfrac{1}{7}\] is a mixed fraction, so we will multiply \[25\times 7\] which gives 175. Now, we will add the number 175 to 1. Thus we have a new number which is 176. Therefore, the mixed fraction changes into a simple fraction. That is we can write \[25\dfrac{1}{7}\] as \[\dfrac{176}{7}\]. By substituting \[25\dfrac{1}{7}\] as \[\dfrac{176}{7}\] we get the volume as \[\dfrac{176}{7}\] and putting it in the equation \[{{h}^{3}}=\dfrac{7}{22}\times 25\dfrac{1}{7}\], we have
\[{{h}^{3}}=\dfrac{7}{22}\times \dfrac{176}{7}\]
\[{{h}^{3}}=8\]
Now, we will divide the power by 3. Therefore, we have
\[{{\left( {{h}^{3}} \right)}^{\dfrac{1}{3}}}={{\left( 8 \right)}^{\dfrac{1}{3}}}\]
\[h={{8}^{\dfrac{1}{3}}}\]
\[h=\sqrt[3]{8}\]
The cube root of 8 is 2. Thus we have h = 2 cm. Hence, option (d) is the correct answer.
Note: While solving such simple questions, students usually make mistakes in units. It should be noted that the unit of the volume here is \[c{{m}^{3}}\] according to the question. If in the question, the volume is in terms of the meter then the volume will be equal to \[{{m}^{3}}\]. Also, note that \[25\dfrac{1}{7}\] is a mixed fraction and one should not make a mistake in writing it into solutions. For example, \[25\dfrac{1}{7}\] is acceptable but \[\dfrac{251}{7}\] is not.