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 __________

Question

Prepp Team · Accepted Answer

To solve this problem, we need to understand the transformation of string P and string K. Let's break down the information:

Initially, string P is laid out as a straight-line segment with length $l$, and string K is a semicircular arc with P as its diameter in Figure (i).
After shortening both strings by length $x$, string K forms a full circle, and string P is its diameter in Figure (ii).

The key points to consider are:

The original length of arc K (semicircle) is $\frac{\pi l}{2}$.
After shortening, K forms a full circle with a new diameter as $(l-x)$. The circumference of this new circle is $\pi (l-x)$.

Since the length of the arc (semicircle) K originally was $\frac{\pi l}{2}$ and it is reduced to form a full circle with length $\pi (l-x)$, we can equate:

$\frac{\pi l}{2} - x = \pi (l-x)$

Simplifying the equation:

$\frac{\pi l}{2} - x = \pi l - \pi x$

Rearranging terms:

$\pi x - x = \pi l - \frac{\pi l}{2}$

$x (\pi - 1) = \frac{\pi l}{2}$

Solving for $x$:

$x = \frac{\pi l}{2(\pi - 1)}$

To find the ratio $\frac{x}{l}$:

$\frac{x}{l} = \frac{\pi}{2(\pi - 1)}$

Therefore, the value of $\frac{x}{l}$ is $\frac{\pi}{2(\pi - 1)}$, which is the correct answer.