Certainly! Let's solve the given problem step-by-step.
We are given:
- Length of the original wire, [tex]\( L_1 = 2725 \)[/tex] feet
- Resistance of the original wire, [tex]\( R_1 = 0.372 \)[/tex] ohms
- Length of the new wire, [tex]\( L_2 = 3600 \)[/tex] feet
We need to find the resistance of the new wire, [tex]\( R_2 \)[/tex].
Given the proportional relationship between resistance and wire length:
[tex]\[
\frac{R_1}{R_2} = \frac{L_1}{L_2}
\][/tex]
We can rearrange this equation to solve for [tex]\( R_2 \)[/tex]:
[tex]\[
R_2 = R_1 \cdot \frac{L_2}{L_1}
\][/tex]
Now, let's substitute the given values into the equation:
[tex]\[
R_2 = 0.372 \cdot \frac{3600}{2725}
\][/tex]
Performing the calculation:
[tex]\[
R_2 = 0.372 \cdot 1.3211009174311927 \approx 0.491
\][/tex]
To the nearest thousandth, the resistance of the 3,600 feet of the same wire is:
[tex]\[
\boxed{0.491} \quad \text{ohms}
\][/tex]
