The velocity for a particle traveling along the x-axis, measured in feet/second, is represented by v(t)=−2((t−3)^2)(t+1)(t−6). At what time(s) does the particle change direction?
Group of answer choices
A. t=1,6
B. t=3,6
C. t=1,3,6
D. t=6
it changes direction when the velocity goes to zero and changes sign, right?
It is clear that v=0 at t=-1,3,6
Since t=3 is a double root, v does not change sign there, so t = -1,6 are the only solutions.
You have a typo, either in the question or the answers.
See the graph at
https://www.wolframalpha.com/input/?i=%E2%88%922%28%28t%E2%88%923%29%5E2%29%28t%2B1%29%28t%E2%88%926%29
To find the times at which the particle changes direction, we need to determine the values of t for which the velocity function, v(t), changes sign.
The velocity equation is v(t) = -2((t-3)^2)(t+1)(t-6)
The particle changes direction when the velocity changes from positive to negative or from negative to positive. This occurs when the velocity function crosses the x-axis.
So, we need to solve the equation v(t) = 0.
Let's set v(t) = 0 and solve for t:
-2((t-3)^2)(t+1)(t-6) = 0
Since the velocity function is a product of factors, we can set each factor equal to zero and solve for t individually.
Setting t-3 = 0, we get t = 3.
Setting t+1 = 0, we get t = -1.
Setting t-6 = 0, we get t = 6.
So, the particle changes direction at t = 3, -1, and 6.
The correct answer choice is C. t=1,3,6.
To determine the time(s) when the particle changes direction, we need to find the values of t for which the velocity changes from negative to positive or from positive to negative.
Given that the velocity function is represented as v(t) = -2((t-3)^2)(t+1)(t-6), we can see that the velocity will change sign when v(t) equals zero. Therefore, we need to solve the equation -2((t-3)^2)(t+1)(t-6) = 0.
To solve this equation, we can set each factor equal to zero and solve for t:
1. Setting t-3 = 0, we find t = 3.
2. Setting t+1 = 0, we find t = -1.
3. Setting t-6 = 0, we find t = 6.
Thus, the possible values of t when the particle changes direction are t = -1, 3, and 6.
The correct answer is C. t=1,3,6.