Answer:
The particle is moving to the left during the interval (-1, 5).
Explanation:
To determine the intervals when the particle is moving to the left, we need to analyze the velocity function v(t) = 3t² - 12t - 15. The particle is moving to the left when its velocity is negative.
First, factor the quadratic expression:
v(t) = 3t² - 12t - 15 = (3t + 3)(t - 5)
The particle's velocity is zero when v(t) = 0, which occurs at t = -1 and t = 5. These points divide the time axis into three intervals: (-∞, -1), (-1, 5), and (5, ∞).
We can evaluate the velocity function at each interval to determine its sign:
Interval (-∞, -1): Choose t = -2
v(-2) = 3(-2)² - 12(-2) - 15 = 12 + 24 - 15 = 21 > 0
Interval (-1, 5): Choose t = 0
v(0) = 3(0)² - 12(0) - 15 = -15 < 0
Interval (5, ∞): Choose t = 6
v(6) = 3(6)² - 12(6) - 15 = 108 - 72 - 15 = 21 > 0
Therefore, the particle is moving to the left during the interval (-1, 5).