Answer: [tex]R = 16.83 \text{ ohms}[/tex]
Explanation:
The resistance of an aluminum wire is given by
[tex]R = \frac{\rho L}{A} [/tex]
where:
[tex]R = \text{ resistance of the wire}
\\ \rho = \text{resistivity of the aluminum} = 2.65 \times 10^{-8} \text{ ohm-meters}
\\ L = \text{length of the wire} = 2 \text{ miles}
\\ A = \text{cross-sectional area of the wire}
[/tex]
Since the resistivity is in ohm-meters, we need to convert the length to meters and the area to square meters. But we need to convert mils to cmils first to get the cross sectional area of the wire.
To convert mils to cmils, we get the square of the diameter of the circle in mils because 1 cmil (or circular mil) is the area of the circle whose diameter is 1 mil. So, since the diameter of the wire is 100 mils, the cross-sectional area of the wire in cmils is given by:
[tex]\text{Area in cmils = }d^2 = 100^2 = 10,000 = 10^4 \text{ cmils}[/tex]
Note that we convert the area in terms of scientific notation because the resistivity is expressed in scientific notation and it is easier to multiply very large and very small numbers if they are expressed in scientific notation.
Now, since [tex]1 \text{ cmil} = 5.067 \times 10^{-10} \text{ square meters}[/tex], the cross sectional area of the wire is equal to
[tex]10^4 \text{ cmils} = (10^4 \text{ cmils} )\left ( \frac{5.067 \times 10^{-10} \text{ square meters}}{1\text{ cmil}} \right ) [/tex]
[tex]10^4 \text{ cmils} = 5.067 \times 10^{-6} \text{ square meters}[/tex]
Since 1 mile = 1,609.344 meters, 2 miles = 2 × 1,609.344 meters = 3,218.688 meters = [tex]3.218688 \times 10^3 \text{ meters}[/tex]. .
Hence the resistance is given by
[tex]R = \frac{\rho L}{A}
\\ R = \frac{(2.65 \times 10^{-8} \text{ ohm-meters}) (3.218688 \times 10^3 \text{ meters})}{5.067 \times 10^{-6} \text{ square meters}}
\\ \boxed{R = 16.83 \text{ ohms}}[/tex]
