Most telephone cables are made of copper wire of either 24 or 26 gauge. If the resistance of 24-gauge wire is 137 Ω/mile and the resistance of 26-gauge wire is 220 Ω/mile, what is the ratio of the diameter of 24-gauge wire to that of 26-gauge wire?

Given that,

Gauge of copper is 24 or 26

The resistance of a wire is given as

R=ρl/A

Where R is resistance on ohms

ρ is resistivity in ohm•m

L is length of wire in m

A is area of cross sectional area m²

R/l = ρ/A

First wire

24 gauge wire has a R/l of 137Ω/mile

Second wire

26gauge wire has a R/l of 220Ω/mile

Both material are made of copper and they both have the same resistivity

R/l = ρ/A

Then,

ρ=RA/l

for 24 gauge

ρ=137A'

A' means area of 24 gauge wire

For 26 gauge

ρ=220A''

A'' means area of 26 gauge wire.

So, the resistivity are equal

Also, the cross section area is given as πr² = πd²/4

Resistivity 24 = Resistivity gauge 26

137A'=220A"

The ratio of diameter gauge 24 to 26 is what we want

137•πd'²/4 = 220•πd"²/4

Then,

137•d'² = 220•d"²

d'²/d"²=220/137

d'²/d"²=1.60584

Take square of both sides

d'/d"=1.267

the ratio of the diameter of 24-gauge wire to that of 26-gauge wire is 1.27 to 3s. f