Question

A group of boys is practising football in a rectangular ground. Raju and Ratan are standing at the two opposite mid-points of the two shorter sides. Raju has the ball, who passes it to Rivu, who is standing somewhere on one of the longer sides. Rivu holds the ball for 3 seconds and passes it to Ratan. Ratan holds the ball for 2 seconds and passes it back to Raju. The path of the ball from Raju to Rivu makes a right angle with the path of the ball from Rivu to Ratan. The speed of the ball, whenever passed, is always 10 metre per second, and the ball always moves on straight lines along the ground.
Consider the following two additional pieces of information:
I. The dimension of the ground is 80 metres × 50 metres.
II. The area of the triangle formed by Raju, Rivu and Ratan is 1000 square metres.
Consider the problem of computing the following: how many seconds does it take for Raju to get the ball back since he passed it to Rivu? Choose the correct option.

• A. I alone is sufficient to solve the problem
• B. II alone is sufficient to solve the problem
• C. Either of I or II, by itself, is sufficient to solve the problem.
• D. I and II both are required to solve the problem.
• E. The problem cannot be solved even with both I and II.

Answer 1

The Correct Option is A

To solve this problem, let's analyze the given information carefully.

1. The dimension of the ground is 80 meters × 50 meters. Therefore, the length of the longer side is 80 meters and that of the shorter side is 50 meters.
2. Raju and Ratan are standing at two opposite mid-points of the shorter sides. This means that both Raju and Ratan are 25 meters away from each of the two longer sides.
3. Rivu is standing somewhere on one of the longer sides such that the path of the ball forms a right angle triangle which means Rivu forms the right angle point.
4. The speed of the ball is 10 meters/second.

Given that Raju passes the ball to Rivu on the longer side. This constitutes one leg of the triangle, say AB.

Rivu then passes the ball to Ratan, forming another leg of the triangle, say BC, which is perpendicular to AB.

The total time taken for Rivu to catch and re-pass the ball is 3 seconds (holding time).

Since the dimensions reveal all necessary lengths considering Riva was on the longer side where he forms a right angle triangle with Raju and Ratan, compute the distance AB + BC using Pythagorean theorem and existing lengths.

The diagonal AC (hypotenuse) is accounted for by two passes: passing time x/10 = 3 + t where t = 2 (time Ratan holds).

The rearrangement and calculation leads only using Option I to understanding Rivan's position.

From this, we can deduce that Ratan will pass the ball back 10 meters/second to the second holder. Thus the return time is consistent and predictable upon current lengths established.

Each step confirms current information sufficiency using I alone.

Answer 2

Let's solve the problem step by step:

1. The rectangular ground has dimensions 80 metres × 50 metres. Raju and Ratan are standing at the midpoints of the shorter sides of the ground, which means they are 50 metres apart from each other since they are on opposite sides of the 50-metre side.
2. Raju passes the ball to Rivu, who is standing somewhere on one of the longer sides (80 metres). Rivu then passes the ball to Ratan, and the path of the ball from Raju to Rivu is perpendicular to the path from Rivu to Ratan.
3. The area of the triangle formed by Raju, Rivu, and Ratan is given as 1000 square metres. However, based on the problem's requirements, we only need to determine if this information is necessary to find the time.
4. The speed of the ball is 10 metres per second.
5. To determine how many seconds it takes for Raju to get the ball back, consider the following:
   • The path Raju to Rivu and Rivu to Ratan forms a right-angled triangle with the perpendicular relationship between the paths. However, given that the ball speed is constant, we can calculate the travel time based on the speed and total distance.
   • Since Raju and Ratan are fixed at midpoints of the opposite sides (50 metres apart), the position of Rivu on the longer side directly affects time calculations.
   • Considering that the information about the area of the triangle is given to emphasize the relationship or positioning of Rivu, it's overly complex since we primarily need the position of Rivu to calculate time. Hence, if we know Rivu's position or the fixed dimensions, we can calculate without the area detail.
6. Thus, using statement I about the ground dimensions, we'd calculate: Raju to Rivu or Rivu to Ratan using constant speed 10 m/s.
7. Rivu holds the ball for 3 seconds, Ratan holds it for 2 seconds.
8. Hence, Raju gets the ball back in the arithmetic time calculated, but for simplicity, the option relies on understanding dimensions affect path, which links to statement I more directly.

Conclusion: I alone is sufficient to solve the problem. Thus, option I alone suffices, making additional area consideration unnecessary to evaluate meaningful duration directly.