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 the semicircle's length as pi times l over 2, and the new circle's circumference as pi times its diameter (l minus x), then equate the shortened K length to that circumference.

Answer 1

The Correct Option is C

Step 1: Write the original length of string K.
String K is a semicircular arc whose diameter equals the length of string P, which is \(l\). The radius of this semicircle is \(l/2\), so the arc length is
\[ K=\pi\times\frac{l}{2}=\frac{\pi l}{2} \]
Step 2: Write the lengths after both strings are shortened by \(x\).
The shortened string P has length \(l-x\), and this becomes the diameter of the new full circle formed by the shortened string K. The shortened string K has length
\[ \frac{\pi l}{2}-x \]
Step 3: Set the shortened K equal to the circumference of the new circle.
A full circle with diameter \(l-x\) has circumference \(\pi(l-x)\). Since the shortened string K forms this circle,
\[ \frac{\pi l}{2}-x=\pi(l-x) \]
Step 4: Solve the equation for \(x\).
\[ \frac{\pi l}{2}-x=\pi l-\pi x \] \[ \pi x-x=\pi l-\frac{\pi l}{2} \] \[ x(\pi-1)=\frac{\pi l}{2} \] \[ x=\frac{\pi l}{2(\pi-1)} \]
Step 5: Find the ratio \(x/l\).
\[ \frac{x}{l}=\frac{\pi}{2(\pi-1)} \]
Step 6: Final Answer.
\[ \boxed{\dfrac{\pi}{2(\pi-1)}} \]
This matches option (C).