SOLUTION: Calculus Question 5. A particle moves according to the law of motion {{{ s=t^3-12t+3, t≥0 }}}, where t is measured in seconds and s in feet. Find the total distance traveled

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Question 1142313: Calculus Question
5. A particle moves according to the law of motion , where t is measured in seconds and s in feet. Find the total distance traveled by the particle in the
first 3 seconds.
(a) 21ft.
(b) 22ft.
(c) 23ft.
(d) 24ft.
(e) None of these are correct
Found 3 solutions by ankor@dixie-net.com, greenestamps, ikleyn:
Answer by ankor@dixie-net.com(22740)   (Show Source): You can put this solution on YOUR website!
Something wrong here, t=3, is not even a positive number!!
Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!




The graph of the function between t=0 and t=3 shows that the function value is 3 and decreasing at t=0, and -6 and increasing at t=3.

That means there is a local minimum between t=0 and t=3.

Calculus shows that minimum (where the derivative is zero) is -13, at t=2. (Or you can use a graphing calculator to find the minimum.)

So the particle travels from 3 to -13 and then from -13 to -6. That gives one of the given answer choices.
Answer by ikleyn(52777)   (Show Source): You can put this solution on YOUR website!
.
The figure below shows the plot of the given function.


And now your (and my) task is to read and to interpret this plot correctly.


From the plot and from the function, you can see that the particle starts from the position s= 3 at t= 0 and first moves
in direction of NEGATIVE s.



    


    Plot y = 



It reaches minimum negative value s= -13 at t= 2; then it changes the direction of moving and moves in direction of POSITIVE s.


     How did I determine it ? - By taking derivative of the function over "t", 
     equating it to zero and finding the turning point moment at t= 2.


Its final position is s= -6 at t= 3.


So, the DISPLACEMENT of the particle from t= 0 to t= 3 is only 3 - (-6) = 9 ft;


but the total way length, or total traveled way is from s= 3 to s= -13 (16 ft) PLUS from -13 to -6 (7 ft).


In all, the traveled distance is 16 + 7 = 23 ft.    ANSWER  (c)


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