Question

A particle moves along the x-axis so that at time t, is greater than or equal to, 0t≥0 its velocity is given by v, left bracket, t, right bracket, equals, 6, t, squared, minus, 84, t, plus, 78, .v(t)=6t 2 −84t+78. Determine all intervals when the particle is moving to the right.

Answer 1

To determine when the particle is moving to the right, we need to find the intervals of time when its velocity is positive. The particle is moving to the right in the intervals (−∞, 6) and (13, +∞).

Step-by-step explanation:

To determine when the particle is moving to the right, we need to find the intervals of time when its velocity is positive. The velocity function is given by v(t) = 6t^2 - 84t + 78. To find the time intervals when the particle is moving to the right, we need to find the solutions to the inequality v(t) > 0.

First, let's find the critical points by setting v(t) = 0:

6t^2 - 84t + 78 = 0

Solving this quadratic equation, we find t = 6 and t = 13. Now we can create a sign chart to determine the sign of v(t) in each interval:

Interval (−∞, 6):

v(t) is positive.

Interval (6, 13):

v(t) is negative.

Interval (13, +∞):

v(t) is positive.

Therefore, the particle is moving to the right in the intervals (−∞, 6) and (13, +∞).