CALCULUS AB
WORKSHEET ON PARTICLE MOTION & OPTIMIZATION
1. A particle moves along a vertical line so that its position at any time is given by
, where s is measured in meters and t in seconds.
(a) When is the particle moving down? Justify your answers.
(b) Find the displacement of the particle after 5 seconds. Show the work that leads to your answer.
(c) Find the distance traveled by the particle between t = 0 and t = 5 seconds. Show the work
that leads to your answer.
(d) When is the particle slowing down? Justify your answer.
2. A particle’s position at time t seconds, , is given by where t is measured in
seconds and s is measured in meters. Find the acceleration of the particle each time the velocity is
zero.
3. A drilling rig 6 miles offshore is to be connected to a refinery on Rig
shore, 18 miles down the coast from the rig by using underwater
pipe from the rig to point and land-based pipe from point P to the
refinery. If underwater pipe costs $50,000 per mile and land-based 6
pipe costs $35,000 per mile, how far should point P be from the
refinery to minimize the cost? What will the cost be? Give the cost P
to the nearest cent. Use Calculus to find and justify your answer. Refinery
4.
t (sec) / 0 / 4 / 7 / 12 / 15meters/sec / 5 / 9 / / /
The velocity of a particle moving along the x-axis is modeled by a differentiable function v, where
the position x is measured in meters. and time t is measured in seconds. Selected values of
are given in the table above.
(a) Use data from the table to estimate the acceleration of the particle at t = 6 seconds. Show the
computations that lead to your answer.
(b) Based on the values in the table, what is the smallest number of instances at which the
velocity could equal 0 on the interval 0 < t < 15? Justify your answer.
5. A cylindrical container has a volume of . Find the radius and height of the cylinder so
that its surface area will be a minimum. Use Calculus to find and to justify your answer.
(Volume of a cylinder = . Surface area of a cylinder = .)