Ankur travels along a rectangular path whose diagonal measures $5\sqrt{13}$ meters. He moves from one end of the diagonal to the other in 5 seconds with a speed less than $4 ext{ m/s}$. What is the length of the longer side of the rectangle?

Question

Ankur travels along a rectangular path whose diagonal measures $5\sqrt{13}$ meters. He moves from one end of the diagonal to the other in 5 seconds with a speed less than $4 	ext{ m/s}$. What is the length of the longer side of the rectangle?

Accepted Answer

**Solution:**

1. Given the diagonal length of the rectangle is $5\sqrt{13}$ meters.

2. Ankur covers this diagonal distance in 5 seconds.

3. Calculate Ankur's speed:
   $$	ext{Speed} = \frac{	ext{Distance}}{	ext{Time}} = \frac{5\sqrt{13}}{5} = \sqrt{13} \approx 3.6 	ext{ m/s}$$

4. Since speed is less than 4 m/s, this speed is valid.

5. Let the sides of the rectangle be $x$ and $y$, where $x > y$.

6. Using the Pythagorean theorem for the diagonal:
   $$x^2 + y^2 = (5\sqrt{13})^2 = 25 	imes 13 = 325$$

7. We need to find integer values of $x$ and $y$ satisfying this equation.

8. Check pairs of integers whose squares sum to 325:
   - $18^2 = 324$, $1^2 = 1$, sum = 325
   - $10^2 = 100$, $15^2 = 225$, sum = 325

9. Since $x > y$, possible pairs are $(18,1)$ or $(15,10)$.

10. The speed is $\sqrt{13} \approx 3.6$ m/s, which is less than 4 m/s, so the distance covered in 5 seconds is $5\sqrt{13}$ meters.

11. The problem asks for the bigger dimension, so among the pairs, the bigger side is 18 meters.

**Answer:** 18 meters

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