The electrical resistance of a wire varies directly with the length of the wire and inversely with the square of the diameter of the wire. If a wire 433 feet long and 4 millimeters in diameter has a resistance of 1 .24 ohms, find the length of a wire whose resistance is 1.49 ohms and whose diameter is 5 millimeters.

Sagot :

Given:

The length of the wire is,

[tex]l=433\text{ feet}[/tex]

The diameter of the wire is,

[tex]d=4\text{ mm}[/tex]

The resistance of the wire is,

[tex]R=1.24\text{ ohms}[/tex]

To find:

The length of the wire whose resistance is,

[tex]R^{\prime}=1.49\text{ ohms}[/tex]

diameter is,

[tex]d^{\prime}=5\text{ mm}[/tex]

Explanation:

The resistance of a conductor is,

[tex]R\propto\frac{l}{d^2}[/tex]

We can write,

[tex]\begin{gathered} \frac{R}{R^{\prime}}=\frac{l}{l^{\prime}}\times\frac{d^{\prime2}}{d^2} \\ \frac{1.24}{1.49}=\frac{433}{l^{\prime}}\times\frac{5^2}{4^2} \\ l^{\prime}=433\times\frac{25}{16}\times\frac{1.49}{1.24} \end{gathered}[/tex]

The length is,

[tex]l^{\prime}=813\text{ feet}[/tex]

Hence, the required length is 813 feet.