Analysis of the problem:

To find the maximum height achieved by the ball, we need to understand the motion of the ball and its key characteristics. The given information states that the ball passes through the same point at 3 seconds and 7 seconds from the start. From this, we can deduce the following:

- The motion of the ball is symmetric, as it passes through the same point at 3 seconds and 7 seconds.
- The total time taken by the ball to reach the maximum height and return to the same point is 7 seconds (from 0 seconds to 7 seconds).
- The time taken by the ball to reach the highest point is half of the total time, which is 3.5 seconds (from 0 seconds to 3.5 seconds).

Finding the maximum height:

To find the maximum height achieved by the ball, we can use the kinematic equation for vertical motion:

h = v₀t - (1/2)gt²

Where:
- h is the height
- v₀ is the initial vertical velocity
- t is the time
- g is the acceleration due to gravity (approximately 9.8 m/s²)

We can use the information given to solve for the maximum height.

Calculations:

1. Finding the initial vertical velocity (v₀):
- At the highest point, the vertical velocity becomes zero. Therefore, we can find the initial vertical velocity by considering the time it takes for the ball to reach the highest point.
- Using the equation for vertical velocity:

v = v₀ - gt

At t = 3.5 seconds:
0 = v₀ - 9.8 * 3.5

Solving the equation, we find v₀ = 34.3 m/s (rounded to one decimal place).

2. Finding the maximum height (h):
- Using the equation for height:

h = v₀t - (1/2)gt²

At t = 3.5 seconds:
h = 34.3 * 3.5 - (1/2) * 9.8 * (3.5)²

Simplifying the equation, we find h = 59.6 meters (rounded to one decimal place).

Conclusion:

Therefore, the maximum height achieved by the ball is approximately 59.6 meters.

125 m if gravity is taken 10 m/s^2