The total distance traveled during the first 6 seconds is 8 feet
How to solve
The Total Distance Traveled
To find the total distance traveled during the first 6 seconds, we need to find the area between the graph of f(t) and the t-axis over the interval [0, 6]. We can do this by breaking the interval into smaller intervals and finding the area under the graph over each interval.
Specifically, let's break the interval [0, 6] into three intervals: [0, 2], [2, 4], and [4, 6]. For each interval, we can find the area under the graph using the trapezoidal rule:
area = (1/2)(b - a)[f(a) + f(b)]
where a and b are the endpoints of the interval.
For the interval [0, 2], we have:
area = (1/2)(2 - 0)[f(0) + f(2)] = (1/2)(2)(0 + 8) = 8
For the interval [2, 4], we have:
area = (1/2)(4 - 2)[f(2) + f(4)] = (1/2)(2)(8 - 0) = 16
For the interval [4, 6], we have:
area = (1/2)(6 - 4)[f(4) + f(6)] = (1/2)(2)(0 - 24) = -24
Adding the areas for all three intervals, we get:
total area = 8 + 16 + (-24) = 8
Therefore, the total distance traveled during the first 6 seconds is 8 feet
The Complete Question
A particle moves according to the law of motion s = f(t), t > 0, where t is measured in seconds and s in feet.
f(t) = t^3 - 9t^2 + 24t
Draw a diagram to illustrate the motion of the particle and use it to find the total distance (in ft) traveled during the first 6 seconds.