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: Write the original lengths of both strings. String P is a straight segment of length \(l\), so its length is simply \(l\). String K is a semicircular arc with string P as its diameter. A semicircle of diameter \(l\) has radius \(l/2\), and the length of a semicircular arc equals \(\pi \times \text{radius}\). So the length of string K is \(\dfrac{\pi l}{2}\).
Step 2: Write the shortened lengths after removing length \(x\) from each string. Shortened string P has length \(l - x\). This shortened P now becomes the diameter of the new circle in Figure (ii). Shortened string K has length \(\dfrac{\pi l}{2} - x\). This shortened K now bends around to form the full circumference of that same new circle.
Step 3: Use the circle relation to connect the two shortened lengths. For a circle, circumference equals \(\pi \times\) diameter. Here the diameter is the shortened P, so the circumference is \(\pi (l - x)\). Since shortened K forms this exact circle, its length must equal this circumference: \[ \frac{\pi l}{2} - x = \pi (l - x) \]
Step 4: Solve the equation for \(x/l\). Expand the right side: \[ \frac{\pi l}{2} - x = \pi l - \pi x \] Bring the \(x\) terms to one side and the \(l\) terms to the other: \[ \pi x - x = \pi l - \frac{\pi l}{2} \] \[ x(\pi - 1) = \frac{\pi l}{2} \] \[ \frac{x}{l} = \frac{\pi}{2(\pi - 1)} \]
Step 5: Check the other options. Option (A), just \(\pi\), ignores the shortening altogether and is really the ratio of the original arc length to the original diameter, not \(x/l\). Option (B), \(\dfrac{\pi - 1}{2\pi}\), is a reciprocal-and-flip of the correct expression, a sign that would come from dividing the equation the wrong way round. Option (D), \(\dfrac{\pi}{\pi - 1}\), drops the factor of 2 that comes from the \(l/2\) term in string K's original length, a common slip.
Final Answer: The value of \(x/l\) works out to \(\dfrac{\pi}{2(\pi - 1)}\), which is option (C). \[ \boxed{\dfrac{x}{l} = \dfrac{\pi}{2(\pi-1)}} \]
