Question

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 ____________

• A. \(\pi\)
• B. \(\dfrac{\pi - 1}{2\pi}\)
• C. \(\dfrac{\pi}{2(\pi - 1)}\)
• D. \(\dfrac{\pi}{\pi - 1}\)

Hint:

Write string K's shortened length as the circumference of the new circle.

Answer 1

The Correct Option is C

Step 1: Write the lengths of P and K before shortening.
String P has length \( l \), and string K is a semicircular arc drawn on P as its diameter, so K's radius is \( l/2 \).
The length of a semicircle is half the circumference of a full circle of that radius, so \( K = \pi \times \dfrac{l}{2} = \dfrac{\pi l}{2} \).

Step 2: Write the lengths after both strings shorten by x.
After shortening, string P has length \( l - x \), and this shortened P becomes the diameter of a full circle formed by the shortened string K.
So the shortened K has length \( \dfrac{\pi l}{2} - x \), and it must equal the circumference of a circle of diameter \( l - x \), which is \( \pi (l - x) \).

Step 3: Solve for x/l.
Setting the two expressions equal: \( \dfrac{\pi l}{2} - x = \pi l - \pi x \).
Collecting the x terms gives \( \pi x - x = \pi l - \dfrac{\pi l}{2} \), so \( x(\pi - 1) = \dfrac{\pi l}{2} \).
This gives \( \dfrac{x}{l} = \dfrac{\pi}{2(\pi - 1)} \).

Final Answer:
The ratio \( x/l \) equals \( \dfrac{\pi}{2(\pi - 1)} \).\[ \boxed{\dfrac{x}{l} = \dfrac{\pi}{2(\pi - 1)}} \]