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Difficulty: MediumDirect and Inverse Proportionality

To study the properties of electrical conductors, a researcher measured the electrical resistance, RR (in ohms, Ω\Omega), of four copper wires. The wires all have the same length but differ in their cross-sectional area, AA (in square millimeters, mm2\text{mm}^2). The measurements are presented in the table below:

WireCross-sectional area, AA (mm2\text{mm}^2)Resistance, RR (Ω\Omega)
10.503.44
21.001.72
32.000.86
44.000.43

Based on the table, if a fifth wire of the same length and material has a cross-sectional area of 8.00 mm28.00\text{ mm}^2, what is its predicted resistance?

  1. 0.215 Ω0.215\ \OmegaAnswer
  2. B
    0.860 Ω0.860\ \Omega
  3. C
    0.000 Ω0.000\ \Omega
  4. D
    3.440 Ω3.440\ \Omega

Answer

0.215 Ω0.215\ \Omega
The correct answer is 0.215 Ω0.215\ \Omega. The data shows that the electrical resistance (RR) is inversely proportional to the cross-sectional area (AA) of the wire, because their product remains constant (A×R=1.72A \times R = 1.72). Therefore, doubling the cross-sectional area from 4.00 mm24.00\text{ mm}^2 to 8.00 mm28.00\text{ mm}^2 requires halving the resistance from 0.43 Ω0.43\ \Omega to 0.215 Ω0.215\ \Omega.

Step-by-Step Solution

1
Determine the mathematical relationship between cross-sectional area (AA) and resistance (RR) from the table.
The product of AA and RR is constant for all trials (0.50×3.44=1.720.50 \times 3.44 = 1.72, 1.00×1.72=1.721.00 \times 1.72 = 1.72, 2.00×0.86=1.722.00 \times 0.86 = 1.72, and 4.00×0.43=1.724.00 \times 0.43 = 1.72). This demonstrates that RR is inversely proportional to AA, with the relationship R=1.72/AR = 1.72 / A.
Identifying whether the relationship is direct or inverse allows us to correctly scale the variables.
2
Calculate the predicted resistance (RR) for a wire with a cross-sectional area of 8.00 mm28.00\text{ mm}^2.
R=1.72/8.00=0.215 ΩR = 1.72 / 8.00 = 0.215\ \Omega. Alternatively, since the area doubles from 4.00 mm24.00\text{ mm}^2 to 8.00 mm28.00\text{ mm}^2, the resistance must be halved: 0.43 Ω/2=0.215 Ω0.43\ \Omega / 2 = 0.215\ \Omega.
Applying the constant of proportionality or the scaling factor determines the final value.

Key Concept

Direct and Inverse Proportionality
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Topics

Direct and Inverse Proportionality

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