Electronics
The electrical resistance (r) of a cable varies directly (y=kx) as its length (l) and inversely (y= k/x) as the square of its diaeter (d). If a cable 16,000 ft long and 1/4 in. in diameter has a resistance of 3.2 ohms, what is the resistance of a cable that is 8000ft long and 1/2 in. in diameter?
Since R = K L/d^2,
where K is a constant, reducing the length L by 1/2 and doubling the diameter d will cause R to decrease by a factor 1/2 x (1/2^2) = 1/8.
What is 1/8 of 3.2 ohms?
To find the resistance of a cable that is 8000 ft long and 1/2 in. in diameter, we can use the given information about the relationship between resistance, length, and diameter.
First, let's calculate the value of the constant K. We know that for the cable with a length of 16,000 ft and a diameter of 1/4 in., the resistance is 3.2 ohms. Using the formula y = kx, we can substitute the values:
3.2 = k * 16000 / (1/4)^2
Simplifying the equation gives us:
3.2 = k * 16000 * 16
Divide both sides of the equation by (16000 * 16) to isolate K:
k = 3.2 / (16000 * 16)
Now we have the value of the constant K, which is approximately equal to 0.0000125.
Next, we need to determine how the resistance changes when we reduce the length by 1/2 and double the diameter. According to the given information, reducing the length L by 1/2 and doubling the diameter d will cause the resistance R to decrease by a factor of 1/8.
Since the initial resistance is 3.2 ohms, we can calculate 1/8 of 3.2 by multiplying it by (1/8).
1/8 * 3.2 = 0.4 ohms
Therefore, the resistance of a cable that is 8000 ft long and 1/2 in. in diameter is 0.4 ohms.