1. The problem involves a quarter circle centered at point S inside rectangle PQRS, where PQ = 35 m and QR = 12 + x m. 2. The arc KL is part of the quarter circle, and we want to find the maximum length of this arc. 3. Since the quarter circle is centered at S, the radius of the quarter circle is the distance from S to the points on the arc KL. 4. The rectangle has sides PQ = 35 m and QR = 12 + x m, so the radius of the quarter circle is the smaller of these two lengths because the quarter circle fits inside the rectangle. 5. Therefore, the radius $r = \min(35, 12 + x)$. 6. To maximize the arc length KL, we want to maximize the radius $r$. 7. Since $x$ is a variable, and $12 + x$ must be less than or equal to 35 for the quarter circle to fit, the maximum radius is when $12 + x = 35$. 8. Solving for $x$, we get $x = 35 - 12 = 23$. 9. The radius $r$ is then $35$ m. 10. The length of the arc KL of a quarter circle is given by $$\text{arc length} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2}.$$ 11. Using $\pi = \frac{22}{7}$ and $r = 35$, we calculate: $$\text{arc length} = \frac{22}{7} \times \frac{35}{2} = \frac{22}{7} \times 17.5 = 22 \times 2.5 = 55.$$ 12. Therefore, the maximum length of the arc KL is $55$ m. 13. Expressed in 4 significant figures, the answer is $55.00$ m.