Consider a string P of length \(l\) that is laid out as a straight-line segment. Another string K is laid out as a semicircular arc with string P as its diameter, as represented in Figure (i). When both the strings are shortened by a length \(x\) they can be re-arranged such that the shortened string K forms a full circle with the shortened string P as its diameter, as represented in Figure (ii). The value of \(x/l\) is ____________
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Write both strings' lengths in terms of \(l\), subtract \(x\) from each, then use the fact that shortened K equals the circumference of the circle whose diameter is shortened P.
Step 1: Set up the problem using the radius of the new circle instead of its diameter. Let $R$ be the radius of the small circle formed in Figure (ii). Since a circle's diameter is twice its radius, and the shortened string P is that diameter: \[ l - x = 2R \quad \Rightarrow \quad x = l - 2R \]
Step 2: Express the arc-length condition using $R$. The original string K (semicircular arc of diameter $l$) has length $\dfrac{\pi l}{2}$. After removing $x$ from it, the leftover length must equal the full circumference of the new circle, which is $2\pi R$: \[ \frac{\pi l}{2} - x = 2\pi R \]
Step 3: Substitute the expression for $x$ from Step 1 into this equation. \[ \frac{\pi l}{2} - (l - 2R) = 2\pi R \] \[ \frac{\pi l}{2} - l + 2R = 2\pi R \] Move all the $R$ terms to one side: \[ \frac{\pi l}{2} - l = 2\pi R - 2R = 2R(\pi - 1) \] \[ R = \frac{l(\pi - 2)}{4(\pi - 1)} \]
Step 4: Go back to $x$ using $x = l - 2R$. \[ x = l - 2 \cdot \frac{l(\pi-2)}{4(\pi-1)} = l - \frac{l(\pi-2)}{2(\pi-1)} \] Write both terms over the common denominator $2(\pi - 1)$: \[ x = \frac{2l(\pi - 1) - l(\pi - 2)}{2(\pi - 1)} = \frac{l[2\pi - 2 - \pi + 2]}{2(\pi - 1)} = \frac{l\pi}{2(\pi-1)} \]
Step 5: Divide by $l$ to get the required ratio. \[ \frac{x}{l} = \frac{\pi}{2(\pi - 1)} \] This confirms the same value reached by working directly with lengths, only this time the algebra was routed through the new circle's radius $R$ as an intermediate variable. Reaching the identical result by a second, independent path is exactly why this answer, option (C), can be trusted. \[ \boxed{\dfrac{x}{l} = \dfrac{\pi}{2(\pi-1)}} \]
