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 ____
Show Hint
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: Understanding the Concept: Since the question only asks for the ratio $x/l$, pick a convenient value for $l$ and solve the numeric case first, then read off the ratio. Take $l = 1$.
Step 2: Key Formula or Approach: With $l = 1$, string P has length $1$ and string K, being a semicircular arc over a diameter of $1$, has length $\dfrac{\pi}{2}$ (half the circumference $\pi \times 1$ of the full circle with that diameter). After cutting a length $x$ from each string, the new string P has length $1 - x$ and the new string K has length $\dfrac{\pi}{2} - x$. The new string K, bent into a full circle, uses the new string P as its diameter, so: \[ \text{new K length} = \pi \times (\text{new P length}) \]
Step 3: Detailed Explanation: Substitute the expressions from Step 2 into that relation: \[ \frac{\pi}{2} - x = \pi(1 - x) \] \[ \frac{\pi}{2} - x = \pi - \pi x \] Move every $x$ term to the left and every constant to the right: \[ \pi x - x = \pi - \frac{\pi}{2} \] \[ x(\pi - 1) = \frac{\pi}{2} \] \[ x = \frac{\pi}{2(\pi - 1)} \] Since $l = 1$ was chosen, this value of $x$ is directly the ratio $x/l$ as well, because $x/l = x/1 = x$. For a general $l$, the same working scales exactly (multiply the string-P length, string-K length, and $x$ all by $l$), so the ratio comes out identical for any choice of $l$.
