Question

2.2 cubic dm of brass is to be drawn into a cylindrical wire of diameter 0.25 cm. Find the approximate length of the wire.

Answer 1

Hint: The volume of the material is given and the diameter of the wire is given. Using the volume of the cylinder formula, \[V = \pi {r^2}h\] , we can calculate the length of the wire.

Complete step-by-step answer:
In the given question, a certain volume of brass is to be drawn into a cylindrical wire of diameter 0.25 cm.
The radius of the cylindrical wire is half of the diameter of the cylindrical wire. Hence, we have:
Radius, \[r = \dfrac{d}{2}\]
\[r = \dfrac{{0.25}}{2}\]
\[r = 0.125cm.........(1)\]
Hence, the radius of the cylindrical wire is 0.125 cm.
When the brass is drawn into a cylindrical wire, the volume remains the same.
It is given that the volume of the brass is 2.2 cubic dm.
We know that 1 dm is 10 cm. Hence, we can convert the volume into cubic centimeter as follows:
Volume, \[V = 2.2d{m^3} = 2.2 \times {10^3}c{m^3}.........(2)\]
We know that formula for the volume of the cylinder. It is given as:
\[V = \pi {r^2}h.........(3)\]
Using equation (1) and equation (2) in equation (3) and the value of \[\pi \] as \[\dfrac{{22}}{7}\], we get:
\[2200 = \dfrac{{22}}{7}{(0.125)^2}h\]
We know that 22 and 2200 cancels and leaves 100 behind in the left-hand side of the equation.
\[100 = \dfrac{1}{7}{(0.125)^2}h\]
Solving for h, we get:
\[h = \dfrac{{100 \times 7}}{{{{\left( {0.125} \right)}^2}}}\]
\[h = 44800cm\]
Expressing the length in meters we get:
\[h = 448m\]
Hence, the length of the cylindrical wire is 448 m.

Note: Always convert all the values to one particular unit and then substitute in the equation for consistency. Otherwise, the result may differ by the powers of 10, resulting in a wrong answer.