To solve this problem using the first method, we need to first find the constant of proportionality in the equation.
Given:
Resistance1 = 9 ohms
Length1 = 50 ft
Diameter1 = 0.05 in
We can substitute these values into the equation and solve for the constant:
9 = constant * 50 / (0.05)^2
9 = constant * 50 / 0.0025
9 = constant * 20,000
Dividing both sides by 20,000, we get:
constant = 0.00045
Now that we have the constant, we can use it to find the resistance in the second scenario where the diameter is 0.02 in.
Given:
Length2 = 50 ft
Diameter2 = 0.02 in
Using the equation, we can substitute the values:
Resistance2 = 0.00045 * 50 / (0.02)^2
Resistance2 = 0.00045 * 50 / 0.0004
Resistance2 = 0.045 * 50
Resistance2 = 2.25 ohms
Therefore, the resistance in 50 ft of a similar wire with a diameter of 0.02 in is 2.25 ohms.
Alternatively, using the second method, we can use the ratio of diameters to find the new resistance.
Given the same values as before:
Diameter1 = 0.05 in
Diameter2 = 0.02 in
We can calculate the ratio of diameters squared:
(diameter2 / diameter1)^2 = (0.02 / 0.05)^2 = (0.4)^2 = 0.16
Now, we can multiply this ratio by the original resistance:
Resistance2 = Resistance1 * (diameter2 / diameter1)^2
Resistance2 = 9 * 0.16
Resistance2 = 1.44 ohms
Using either method, we find that the resistance in 50 ft of a similar wire with a diameter of 0.02 in is 2.25 ohms or 1.44 ohms, respectively.