Each of four particles moves along an x axis.Their coordinates (in meters)as functions of time (in seconds)are given by For which of these particles is the velocity increasing for t 0?
A)All four
B)Only 1
C)Only 2 and 3
D)Only 2, 3, and 4
E)None of them

Question

Each of four particles moves along an x axis.Their coordinates (in meters)as functions of time (in seconds)are given by  For which of these particles is the velocity increasing for t > 0?
A)All four 
B)Only 1 
C)Only 2 and 3 
D)Only 2, 3, and 4 
E)None of them

Quizplus · Accepted Answer

The velocity of a particle is given by its derivative with respect to time.

Particle 1: v(t) = -3t² + 24t - 27
v'(t) = -6t + 24

v'(t) is decreasing for all t > 0, so the velocity of particle 1 is not increasing.

Particle 2: v(t) = 9t - 21
v'(t) = 9

v'(t) is constant and positive for all t > 0, so the velocity of particle 2 is increasing.

Particle 3: v(t) = 6t² - 48t + 87
v'(t) = 12t - 48

v'(t) is increasing for t > 4, so the velocity of particle 3 is increasing for t > 4.

Particle 4: v(t) = -2t³ + 24t² - 102t + 143
v'(t) = -6t² + 48t - 102

v'(t) is a quadratic function with a negative leading coefficient, so it has a maximum value at t = 4. Since v'(4) = 6, which is positive, and v'(t) is decreasing for t > 4, the velocity of particle 4 is increasing for t in the interval (0,4).

Therefore, the only particles whose velocity is increasing for t > 0 are particle 2 and particle 3. The correct answer is choice C, "Only 2 and 3."

Each of Four Particles Moves Along an X Axis