The position of a moving particle is given by the position function f(t) = 6t t2
ID: 2891999 • Letter: T
Question
The position of a moving particle is given by the position function f(t) = 6t t2 0.3t3 + 0.1t4, 0 t 8. Estimate values from the graphs to answer the following questions.
(a) At what time does the particle reverse direction? (Round your answer to one decimal place.) t =
(b) When is the displacement positive? (Round your answer to one decimal place. Enter your answer using interval notation.) When is the displacement negative? (Round your answer to one decimal place. Enter your answer using interval notation.)
(c) When is the particle's acceleration positive? (Round your answer to one decimal place. Enter your answer using interval notation.) When is the particle's acceleration negative? (Round your answer to one decimal place. Enter your answer using interval notation.)
(d) Write equations for the velocity and acceleration functions.
f'(t) =
f''(t) =
Explanation / Answer
From the given question,
position function:
f(t) = 6t t2 0.3t3 + 0.1t4, 0 t 8.
velocity function:
f '(t)= -6 -2t -0.9t2 + 0.4t3
acceleration function:
f ''(t)= -2 - 1.8 t + 1.2t2
(a) particle reverses the direction when velocity is zero.
-6 -2t -0.9t2 + 0.4t3 =0
It has one real solution, t=4.25sec
so particle reverses direction after t=4.25sec.
b) to know displacement negative, we have to find time when f(x)=0
roots are t=0 and t=6.18sec
displacement is positive 0<t<6.18
displacement is negative 6.08<t<8
c) acceletion is positive when f ''(t) >0
-2 - 1.8 t + 1.2t2 >0
t >2.24
acceleration is positive t>2.24
acceleration is negative 0<t<2.24
d)velocity function:
f '(t)= -6 -2t -0.9t2 + 0.4t3
acceleration function:
f ''(t)= -2 - 1.8 t + 1.2t2
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