Answer:

Explanation:
Given that, the range covered by the sphere,  , when released by the robot from the height,
, when released by the robot from the height,  , with the horizontal speed
, with the horizontal speed  is
 is  as shown in the figure.
 as shown in the figure.
The initial velocity in the vertical direction is  .
.
Let  be the acceleration due to gravity, which always acts vertically downwards, so, it will not change the horizontal direction of the speed, i.e.
 be the acceleration due to gravity, which always acts vertically downwards, so, it will not change the horizontal direction of the speed, i.e.  will remain constant throughout the projectile motion.
 will remain constant throughout the projectile motion.
So, if the time of flight is  , then
, then

Now, from the equation of motion

Where  is the displacement is the direction of force,
 is the displacement is the direction of force,  is the initial velocity,
 is the initial velocity,  is the constant acceleration and
 is the constant acceleration and  is time.
 is time.
Here,  and
 and  (negative sign is for taking the sigh convention positive in
 (negative sign is for taking the sigh convention positive in  direction as shown in the figure.)
 direction as shown in the figure.)
So, from equation (ii),



Similarly, for the launched height  , the new time of flight,
, the new time of flight,  , is
, is

From equation (iii), we have

Now, the spheres may be launched at speed  or
 or  .
.
Let, the distance covered in the  direction be
direction be  for
 for  and
 and  for
 for  , we have
, we have

 [from equation (iv)]
 [from equation (iv)]
 [from equation (i)]
 [from equation (i)]
 (approximately)
 (approximately)
This is in the  points range as given in the figure.
 points range as given in the figure.
Similarly, 
 [from equation (iv)]
 [from equation (iv)]
 [from equation (i)]
 [from equation (i)]
 (approximately)
 (approximately)
This is out of range, so there is no point for  .
.
Hence, students must choose the speed  to launch the sphere to get the maximum number of points.
 to launch the sphere to get the maximum number of points.