Question

Diameter of cylinder A is 7cm, and the height is 14cm. Diameter of cylinder B is 14cm and height is 7cm. Without doing any calculations, can you suggest whose volume is greater? Verify it by finding the volume of both the cylinders. Check whether the cylinder with greater volume also has the greater surface area.

Answer 1

Hint: In this question, we should try to find out the dependencies of volume on its height and diameter. In the cylinders, the diameter of the first is half of that of the second whereas the height of the first is double than the second. Therefore, using the formula for volume we can find the dependency of volume on height and diameter and obtain the answer. The surface area can be found from its formula.

Complete step-by-step answer:

We know that the diameter(d) of a cylinder should be twice its radius(r). Thus,

$d=2r...........(1.1)$

Now, the volume of a cylinder of radius r and height h is given by the formula

$V=\pi {{r}^{2}}h=\dfrac{1}{2}\pi {{d}^{2}}h..........(1.2)$

Therefore, we see that the volume depends on the square of the r and just to the first power of h…...... (1.3)

Now, in the question, it is given that the diameter of the first cylinder is half of that of the second whereas the height of the first cylinder is double than that of the second. Therefore, from equation (1.3), we find that as the dependency on diameter $\left( V\propto {{r}^{2}} \right)$ is more than the dependency on height $\left( V\propto h \right)$, the volume of the second cylinder should be more.

By explicit calculation, if we take the volume, diameter and height of the first cylinder to be ${{V}_{1}},{{d}_{1}},{{h}_{1}}$ and that of the second cylinder to be ${{V}_{2}},{{d}_{2}},{{h}_{2}}$ respectively, then from the question, it is given that

$\begin{align}

  & {{d}_{1}}=7cm,{{h}_{1}}=14cm \\

 & {{d}_{2}}=14cm,{{h}_{2}}=7cm..............(1.3) \\

\end{align}$

Therefore, using the formula from equation (1.2), we get

 $\begin{align}

  & \dfrac{{{V}_{1}}}{{{V}_{2}}}=\dfrac{\dfrac{1}{2}\pi {{d}_{1}}^{2}{{h}_{1}}}{\dfrac{1}{2}\pi {{d}_{2}}^{2}{{h}_{2}}}=\dfrac{\dfrac{1}{2}\pi \times {{7}^{2}}\times 14}{\dfrac{1}{2}\pi \times {{14}^{2}}\times 7}=\dfrac{7}{14}=0.5<1 \\

 & \Rightarrow {{V}_{1}}<{{V}_{2}}..........(1.4) \\

\end{align}$

Thus, we have found that the volume of the first cylinder is less than the volume of the second cylinder by explicit calculation.


Now, the formula for surface area is given by

$S=2\pi rh=\pi dh\text{ (Using equation 1}\text{.1)}...........\text{(1}\text{.5)}$

Thus, if we denote the surface area of the first and second cylinders by ${{S}_{1}}$ and ${{S}_{2}}$, then using the values given in (1.3), we get

$\begin{align}

  & \dfrac{{{S}_{1}}}{{{S}_{2}}}=\dfrac{\pi {{d}_{1}}{{h}_{1}}}{\pi {{d}_{2}}{{h}_{2}}}=\dfrac{7\times 14}{14\times 7}=1 \\

 & \Rightarrow {{S}_{1}}={{S}_{2}}.....................(1.6) \\

\end{align}$

Thus, the surface area of the two cylinders are equal.

Note: In equations (1.4) and (1.6), we have not written the units of the diameter and height because as it is a ratio, the units from the numerator and denominator cancel out. However, while considering an equation where the units will not cancel out, we should calculate the quantities with their units.