Final answer:
To calculate the total distance traveled, find the velocity function by differentiating the position function, solve for times when the velocity is zero to check for direction changes, and then integrate the absolute value of the velocity function over the given time interval.
Step-by-step explanation:
To find the total distance traveled by a body given the position function S = t^3 - 12t^2 + 45t from t=0 to t=4, we first need to determine the velocity function by taking the derivative of S with respect to time. The velocity function v(t) = dS/dt is the derivative of the position function, so v(t) = 3t^2 - 24t + 45.
However, the question is asking for the total distance traveled, not just the displacement. To find this, we must consider the changes in the direction of motion. This involves finding the times at which the velocity is zero, which may indicate a change in direction. Setting the velocity function equal to zero gives the times t when the body changes direction: 3t^2 - 24t + 45 = 0. By solving this quadratic equation, we can find such points, but since we are only considering the interval from t=0 to t=4, we'll check which of these points fall within this interval.
Once we have the times where the body changes direction, we calculate the distance traveled between these points by integrating the absolute value of the velocity function. The reason we use the absolute value is to ensure that we are adding up distances traveled in all directions, without subtracting any distance when the velocity might be negative. The sum of these distances will give us the total distance traveled by the body during the given time interval.