Question

Find the length of 11 kg copper wire of diameter 0.4 cm. Given one cubic cm of copper weighs 8.4 gm.

Answer 1

Hint: Recall the definition of density. Use the formula for density to find the volume of the 11kg copper wire. Use the volume formula of a cylinder to find the length of the copper wire.

Complete step-by-step answer:
Density is a measurement that compares the amount of matter an object has to its volume. It is the ratio of the mass of an object to its volume.
Density = \[\dfrac{{Mass}}{{Volume}}..............(1)\]
It is given that the density of the copper wire is 8.4 gm per cubic cm.
We convert it into kg per cubic cm by dividing by 1000.
\[8.4gm/c{m^3} = 8.4 \times {10^{ - 3}}kg/c{m^3}...........(2)\]
The mass of the copper wire is 11 kg.
Let the volume of the copper wire be V, then using equation (2) and equation (1), we have:
\[8.4 \times {10^{ - 3}} = \dfrac{{11}}{V}\]
Solving for V, we have:
\[V = \dfrac{{11}}{{8.4 \times {{10}^{ - 3}}}}\]
\[V = \dfrac{{110000}}{{84}}c{m^3}...........(3)\]

We know the formula for the volume of a cylinder of radius ‘r’ and length ‘h’ is given as follows:
\[V = \pi {r^2}h............(4)\]
It is given that the diameter of the wire is 0.4 cm, then the radius is half of it.
\[r = \dfrac{{0.4}}{2}\]
\[r = 0.2cm...........(5)\]
Substituting equation (5) in equation (4), we have:
\[V = \pi {(0.2)^2}h\]
\[V = 0.04\pi h............(6)\]
Equating equation (6) to equation (3), we get:
\[0.04\left( {\dfrac{{22}}{7}} \right)h = \dfrac{{110000}}{{84}}\]
Solving for h, we have:
\[h = \dfrac{{110000}}{{84}}\left( {\dfrac{7}{{22 \times 0.04}}} \right)\]
\[h = \dfrac{{110000}}{{84}}\left( {\dfrac{7}{{22 \times 0.04}}} \right)\]
\[h = 10416.66cm\]
Converting it into meters, we have:
\[h = 104.17m\]
Hence, the length of the copper wire is \[104.17m\].

Note: Convert all quantities to the same units before adding, multiplying, or dividing them to get the correct answer, otherwise, your answer will differ by powers of 10 from the correct answer, which will be wrong.