Two particles move along the x-axis. For 0≤t≤6, the position of particle P is p(t)=2cos(t*π/4), while the position of particle R is r(t)=t^3-6t^2+9t+3.
For 0≤t≤6 find all the times which the two particles travel in opposite directions.
when is -pi/2 sin(t*pi/4) < 0?
when is sinx > 0?
for 0 < x < pi
so, we need 0 < t*pi/4 < pi
0 < t < 4
in that interval, sin(t*pi/4) > 0, so -pi/2 sin(t*pi/4) < 0
similarly, dq/dt > 0 for t<1 or t>3
so, on 0<t<1 and 3<t<4
dp/dt < 0 and dq/dt > 0, so P and Q are moving in opposite directions.
solve the other way to see when P is moving right and Q is moving left.
don't guess and check. solve the inequalities. That's just algebra, not calculus.
P is moving to the right when dp/dt > 0
R is moving to the left when dr/dt < 0
dp/dt = -pi/2 sin(t*pi/4)
dq/dt = 3t^2 - 12t + 9 = 3(t^2-4t+3) = 3(t-1)(t-3)
now it should be fairly easy. what do you get?
what values do i plug in for t? do i have to guess and check?
To find the times at which the two particles travel in opposite directions, we need to determine when the velocity of one particle is positive and the velocity of the other particle is negative.
First, let's find the velocity of particle P. The velocity is the derivative of the position function p(t). So, we differentiate p(t) with respect to t:
p'(t) = -2sin(t*π/4) * (π/4)
Next, let's find the velocity of particle R. The velocity is the derivative of the position function r(t). So, we differentiate r(t) with respect to t:
r'(t) = 3t^2 - 12t + 9
Now, we need to find the times at which the velocities have opposite signs. In other words, we need to find the values of t for which p'(t) and r'(t) have opposite signs.
To do this, we can set up the following inequality:
p'(t) * r'(t) < 0
Simplifying the inequality, we have:
(-2sin(t*π/4) * (π/4)) * (3t^2 - 12t + 9) < 0
Now, we can solve this inequality to find the values of t for which the two particles travel in opposite directions.
Alternatively, we can plot the graphs of p'(t) and r'(t) on a graphing calculator or software and observe the points where the graphs intersect the t-axis (x-axis). These points correspond to the times when the particles travel in opposite directions.
Note: For the given time interval of 0 ≤ t ≤ 6, using a graphing calculator or software may be more convenient in visually identifying the points at which the particles travel in opposite directions.