A certain copper wire has a resistance of 12.0 \Omega.At what point along its length must the wire be cut so that the resistance of one piece is 5.0 times the resistance of the other?

Let's assume that the original length of the copper wire is "L".

To find the point at which the wire should be cut, we need to determine the lengths of the two resulting pieces.

Let "x" represent the length of the first piece and "L - x" represent the length of the second piece.

The resistance of a wire is directly proportional to its length, so we can set up the following equation:

x / (L - x) = 5

The resistance of the first piece (x) divided by the resistance of the second piece (L - x) is equal to 5.

Now, let's solve this equation algebraically:

Cross multiply to obtain:

x = 5(L - x)

Expand:

x = 5L - 5x

Now, isolate "x" on one side of the equation:

x + 5x = 5L

Combine like terms:

6x = 5L

Divide both sides by 6:

x = 5L / 6

Therefore, the wire should be cut at a point that is 5/6 (or 0.83) times the original length of the wire.

To find the point along the length of the wire where it should be cut, such that the resistance of one piece is 5.0 times the resistance of the other, we can use the formula for the resistance of a wire in series.

The resistance of a wire is directly proportional to its length. Therefore, if we cut the wire at some point, the resistance of one piece will be directly proportional to the length of that piece, and the resistance of the other piece will be directly proportional to the length of the other piece.

Let's assume that the wire is cut at a distance x from one end, so the length of one piece of the wire is x, and the length of the other piece is L - x, where L is the total length of the wire.

According to the problem, the resistance of one piece should be 5.0 times the resistance of the other piece. Therefore, we can write the equation as:

Resistance of one piece = 5 * Resistance of other piece

Using the formula for the resistance of a wire in series, R = ρ * (L/A), where R is the resistance, ρ is the resistivity of the material (a constant), L is the length of the wire, and A is the cross-sectional area of the wire (also a constant):

R1 = 5 * R2

ρ * (x / A) = 5 * ρ * ((L - x) / A)

Cross-multiplying and simplifying the equation, we get:

x = 5 * (L - x)

Solving for x, we have:

6x = 5L

x = (5L) / 6

Therefore, to find the point along the length of the wire where it should be cut, we can multiply the total length of the wire by (5/6).

resistance is proportional to length, so the lengths have to be in the ratio of 1:4, so the cut must be at 1/5 the length, giving you lengths of 1/5 and 4/5 the length. Actual resistances are 10/5 ohms and 40/5 ohms, or R1=2 ohms and R2=8