Understanding the problem:
We are given that a ball is thrown vertically upwards with an initial velocity of 19.6 m/s and it returns in 4 seconds. We need to find the maximum height attained by the ball.

Key concepts:
1. When a ball is thrown upwards, its initial velocity is positive and its final velocity is zero at its highest point.
2. The acceleration due to gravity acts in the opposite direction to the motion of the ball, causing it to slow down as it moves upwards and speed up as it moves downwards.
3. The time taken for the ball to reach its highest point is equal to the time taken for it to return to its initial position.

Step-by-step solution:
Let's solve the problem step-by-step:

Step 1: Calculate the time taken for the ball to reach its highest point.
Since the ball returns in 4 seconds, the time taken for it to reach its highest point is half of that, i.e., 4/2 = 2 seconds.

Step 2: Calculate the final velocity of the ball at its highest point.
Using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can calculate the final velocity of the ball at its highest point.
Given:
Initial velocity (u) = 19.6 m/s (upwards)
Acceleration (a) = -9.8 m/s^2 (due to gravity)

v = u + at
v = 19.6 + (-9.8)(2)
v = 19.6 - 19.6
v = 0 m/s

Therefore, the final velocity of the ball at its highest point is 0 m/s.

Step 3: Calculate the maximum height attained by the ball.
Using the equation v^2 = u^2 + 2as, where s is the displacement, we can calculate the maximum height attained by the ball.
Given:
Initial velocity (u) = 19.6 m/s (upwards)
Final velocity (v) = 0 m/s
Acceleration (a) = -9.8 m/s^2 (due to gravity)

v^2 = u^2 + 2as
0^2 = 19.6^2 + 2(-9.8)s
0 = 384.16 - 19.6s
19.6s = 384.16
s = 384.16/19.6
s ≈ 19.6 m

Therefore, the maximum height attained by the ball is approximately 19.6 meters.

Conclusion:
The correct answer is option 'C' - 19.6 m.