Question

A metal pipe has a bore (inner diameter) of 5 cm. The pipe is 5 mm thick all round. Find the weight, in kilograms, of 2 meters of the pipe if $1\text{ c}{{\text{m}}^{3}}$ of the metal weighs 7.7 grams.
(a) 12.31 kg
(b) 19.31 kg
(c) 13.31 kg
(d) 14.31 kg

Answer 1

Hint:
 The shape of the pipe is a cylinder. We know the thickness of the pipe, so we will calculate the inner radius of the pipe and the outer radius of the pipe. Then we will use the formula for the volume of the cylinder to find the quantity of metal in the pipe with height 2 meters. After that, we will find the weight of the pipe by using the information that $1\text{ c}{{\text{m}}^{3}}$ of the metal weighs 7.7 grams.

Complete step by step answer:
The shape of the pipe is a cylinder. The inner diameter of the pipe is given as 5 cm. Therefore, the inner radius of the pipe is 2.5 cm. We know that the pipe is 5 mm thick all around. We know that $\text{5 mm}=0.5\text{ cm}$. So, the outer radius of the pipe will be $2.5+0.5=3\text{ cm}$. Let us draw a rough diagram to understand the shape of the pipe and its dimensions. The figure looks like the following,

To find the quantity of metal in the pipe, we will find the volume of the outer cylinder and the volume of the inner cylinder. And then we will subtract the volume of the inner cylinder from the volume of the outer cylinder. The volume of the cylinder is given by
$\text{volume of cylinder}=\pi {{r}^{2}}h$
where $r$ is the radius of the cylinder and $h$ is the height of the cylinder.
We are given that the height of the pipe is 2 meters, which is 200 cm.
Let us find the volume of the outer cylinder.
$\begin{align}
  & {{V}_{outer}}=\pi \times {{\left( 3 \right)}^{2}}\times 200 \\
 & \therefore {{V}_{outer}}=1800\pi \text{ c}{{\text{m}}^{3}} \\
\end{align}$
Now, using the same formula, we will find the volume of the inner cylinder.
$\begin{align}
  & {{V}_{inner}}=\pi \times {{\left( 2.5 \right)}^{2}}\times 200 \\
 & \therefore {{V}_{inner}}=1250\pi \text{ c}{{\text{m}}^{3}} \\
\end{align}$
Therefore, the quantity of the metal in the pipe can be calculated as follows,
\[\begin{align}
  & \text{quantity of metal}={{V}_{outer}}-{{V}_{inner}} \\
 & \Rightarrow \text{quantity of metal}=1800\pi -1250\pi \\
 & \therefore \text{quantity of metal}=550\pi \text{ c}{{\text{m}}^{3}} \\
\end{align}\]
Now, we know that $1\text{ c}{{\text{m}}^{3}}$ of the metal weighs 7.7 grams. So, the quantity of metal in grams is,
$\text{quantity of metal}=550\pi \times 7.7$
Substituting $\pi =\dfrac{22}{7}$ in the above equation, we get
$\begin{align}
  & \text{quantity of metal}=550\times \dfrac{22}{7}\times 7.7 \\
 & \Rightarrow \text{quantity of metal}=550\times 22\times 1.1 \\
 & \therefore \text{quantity of metal}=13310\text{ grams} \\
\end{align}$
We know that $1\text{ kg = 1000 grams}$. Therefore, $13310\text{ grams = 13}\text{.31 kg}$. Hence, the correct option is (c).
Note:
It is always useful to draw a rough diagram to understand the shape and dimensions of the given object. This helps us in choosing the correct formula. It is essential that we are familiar with the formulae for volume and areas of the standard geometric objects. We should do the calculations explicitly so that we can avoid making any minor errors and obtain the correct answer.