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 string K's length as half the circumference of a circle whose diameter is string P, then equate the shortened K to the circumference of a circle whose diameter is the shortened P.
Step 1: Find the length of string K. String P has length \(l\) and it forms the diameter of the semicircular arc bent from string K, as shown in Figure (i). A full circle with diameter \(l\) has circumference \(\pi l\). A semicircular arc of the same diameter is exactly half of that circle's boundary, so its length is \(\dfrac{\pi l}{2}\). So the length of string K is \(\dfrac{\pi l}{2}\).
Step 2: Write the new lengths after both strings are shortened. Each string loses the same length \(x\) when it is shortened. The shortened string P has length \(l - x\), and Figure (ii) shows this shortened string P acting as the diameter of a full circle formed by the shortened string K. The shortened string K has length \(\dfrac{\pi l}{2} - x\).
Step 3: Set up the circle equation. Since the shortened string K is bent into a complete circle whose diameter is the shortened string P, the length of the shortened string K must equal the circumference of that circle. \[ \frac{\pi l}{2} - x = \pi(l - x) \]
Step 4: Solve for the ratio \(x/l\). Expand the right side of the equation: \[ \frac{\pi l}{2} - x = \pi l - \pi x \] Collect the \(x\) terms on one side and the \(l\) terms on the other: \[ \pi x - x = \pi l - \frac{\pi l}{2} \] \[ x(\pi - 1) = \frac{\pi l}{2} \] Divide both sides by \(l(\pi - 1)\): \[ \frac{x}{l} = \frac{\pi}{2(\pi - 1)} \]
Step 5: Check why the other options are wrong. Option (A), \(\pi\), comes up if someone sets the shortened K length equal to \(\pi l\) instead of \(\pi(l-x)\), forgetting that the new circle's diameter is the shortened P, not the original P. Option (B), \(\dfrac{\pi-1}{2\pi}\), is the reciprocal-looking flip of the correct ratio, picked up by inverting the equation at the wrong step. Option (D), \(\dfrac{\pi}{\pi-1}\), appears if the factor of \(\dfrac{1}{2}\) from the semicircle length is dropped, effectively treating string K's original length as \(\pi l\) instead of \(\dfrac{\pi l}{2}\).
Final Answer: The value of \(x/l\) is \(\dfrac{\pi}{2(\pi - 1)}\), which is option (C). \[ \boxed{\dfrac{x}{l} = \dfrac{\pi}{2(\pi-1)}} \]
