The angle at which the range of a projectile is maximized depends on the initial velocity and the acceleration due to gravity, but it is independent of the height of the tower. The maximum range is achieved when the projectile is launched at an angle of 45 degrees with respect to the horizontal.
To understand why this is the case, let's consider the motion of a projectile. When a ball is thrown from a tower at an angle θ, it follows a curved trajectory called a parabola. The vertical and horizontal components of its motion are independent of each other.
The horizontal component of the initial velocity determines how far the ball will travel horizontally (the range), while the vertical component determines the height and time of flight. For a given initial velocity, if you launch the ball at an angle greater than 45 degrees, the vertical component becomes larger, causing the ball to spend more time in the air and reach a greater height but sacrificing some of the horizontal distance. Conversely, launching the ball at an angle less than 45 degrees results in less time in the air and a smaller height but sacrifices horizontal distance as well.
At an angle of 45 degrees, the initial velocity is split equally between the horizontal and vertical components, optimizing the range. This can be proven mathematically using projectile motion equations and calculus, but conceptually it is because the ball spends an equal amount of time going up and coming down, maximizing the time of flight, which in turn maximizes the horizontal distance covered.
Therefore, if you want to achieve the maximum range when throwing a ball from a tower, the optimal angle to launch it is 45 degrees with respect to the horizontal.