A 19-foot piece of copper wire produces a resistance of 0.03 ohm. Use the constant of proportionality
k = 5.73 × 10−8
ohm · foot to find the diameter of the wire. (Convert to inches and round your answer to three decimal places.)
To find the diameter of the wire, we can use the formula for resistance of a wire:
R = (k * L) / (A)
where:
R is the resistance,
k is the constant of proportionality,
L is the length of the wire, and
A is the cross-sectional area of the wire.
Here, we are given:
R = 0.03 ohm
k = 5.73 × 10^(-8) ohm · foot
L = 19 feet
Since we need to find the diameter of the wire, we can express the cross-sectional area A in terms of the radius r:
A = π * r^2
Now, let's solve the given equation for A, in terms of r:
R = (k * L) / (A)
A = (k * L) / R
π * r^2 = (k * L) / R
r^2 = (k * L) / (R * π)
To find the radius, we take the square root of both sides:
r = √((k * L) / (R * π))
Now, let's plug in the given values and solve:
r = √((5.73 × 10^(-8) ohm · foot * 19 feet) / (0.03 ohm * π))
r = √((1.08927 × 10^(-6) ohm · foot^2) / (0.03 ohm * π))
r ≈ √(3.6309 × 10^(-5) foot^2)
r ≈ 0.1904 feet
To convert the diameter to inches, we multiply by the conversion factor:
1 foot = 12 inches
So, the diameter in inches would be:
d = 2 * r * 12
d ≈ 2 * 0.1904 * 12
d ≈ 4.5688 inches
Rounding to three decimal places, the diameter of the wire is approximately 4.569 inches.