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: Express everything as a fraction of $l$.
Let $x=kl$, where $k=x/l$ is what we want to find. The original semicircle K has length $\dfrac{\pi l}{2}$, since its diameter is $l$.

Step 2: Write both shortened lengths using $k$.
Shortened P, the new diameter, is $l-kl=l(1-k)$. Shortened K is $\dfrac{\pi l}{2}-kl=l\left(\dfrac{\pi}{2}-k\right)$.

Step 3: The shortened K must trace a full circle on the shortened P.
A circle with diameter $l(1-k)$ has circumference $\pi l(1-k)$. Setting this equal to the shortened K length,
\[ l\left(\frac{\pi}{2}-k\right)=\pi l(1-k) \]

Step 4: Cancel $l$ and solve for $k$.
\[ \frac{\pi}{2}-k=\pi-\pi k \]
\[ \pi k - k = \pi - \frac{\pi}{2} \]
\[ k(\pi-1) = \frac{\pi}{2} \]
\[ k=\frac{\pi}{2(\pi-1)} \]

Step 5: Conclude.
Since $k=x/l$,
\[ \boxed{\dfrac{x}{l}=\dfrac{\pi}{2(\pi-1)}} \]