Final answer:
To answer the question, one must find when the box stops by integrating acceleration obtained from the time-dependent force. After finding the time it takes for the box to stop, the total displacement and velocity at 3.50 seconds can be calculated using calculus.
Step-by-step explanation:
To determine the distance the box moves before coming to rest and its velocity at 3.50 seconds, we need to solve two different problems. First, we need to find out when the box stops, which also means finding the time when its velocity becomes zero. We can then use that time to calculate the distance.
The force applied is a time-dependent force given by f(t) = (6.00 N/s²)t². Since the force increases with time, acceleration is not constant, and thus we cannot use the simple kinematic equations directly. Instead, we apply Newton's second law to find acceleration at any time t, which is a(t) = f(t)/m, and then integrate this acceleration to find velocity as a function of time and subsequently the position as a function of time.
Once we know when the box comes to rest, we use calculus to find the total displacement during that time. If we assume the force continues to be applied past the point where the box stops, we would then calculate the velocity at 3.50 seconds using the velocity-time relationship we derived from the integration of acceleration.
Due to the complexity of this problem involving the integration of a non-constant force, students requiring step-by-step integration and solution should consult a textbook or ask for further detailed guidance from their teachers.