Extension inversely proportional to cross-sectional area

Steps:

• Hooke's law: extension $e = \frac{F L}{A Y}$, where $A$ is area, $Y$ is Young's modulus (same for both), $L$ is length (same).
• Graph slope $\frac{de}{dF} = \frac{L}{A Y}$, so slope .
• Area $A \propto d^2$, so slope $\propto \frac{1}{d^2}$.
• If slope of Q is $\frac{1}{3}$ slope of P (from graph), then $\frac{d_Q^2}{d_P^2} = 3$, so .

Why C is correct:

• Slope ratio 1:3 implies area ratio 3:1, and since area $\propto d^2$, diameter ratio is $\sqrt{3}$ by definition of proportional areas.

Why the others are wrong:

• A: Linear diameter ratio ignores $d^2$ dependence of area.
• B: $1/\sqrt{3}$ would mean Q thinner than P, reversing slope ratio.
• D: Linear ratio 3 would imply area ratio 9, yielding slope ratio 1:9.

Final answer: C