1. The position of a particle along a straight line is given by s = (2.5t^3-13.5t^2+22.5t) ft, where t is in seconds. Determine the position of the particle when t = 6 s and the total distance it travels during the 6-s time interval. Hint: Plot the path to determine the total distance traveled.

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## 1 Answer

s(t)=2.5t³-13.5t²+22.5t, s(6)=189 ft, making the final position (6,189).

s(0)=0, so the distance travelled in a straight line is the same as the position=189 ft. However, the actual distance travelled along the curved path has to be calculated from the arc length, x. This will create a line integral.

We can write, using Pythagoras:

dx²=dt²+ds², so (dx/dt)²=1+(ds/dt)².

ds/dt=7.5t²-27t+22.5=3(2.5t²-9t+7.5).

dx/dt=√[1+9(2.5t²-9t+7.5)²].

x=∫₀⁶√[1+9(2.5t²-9t+7.5)²]dt.

This is not easy to integrate but approximations for the definite integral are available by dividing the interval [0,6] into small segments. I think this method might be beyond the scope of this problem.

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