two particles are moving along a linear track according to the functions x1=t^2+2t and x2=-t^2+8t+80 for t is greater than or equal to zero.

a)at what value of t will the two particles be in the same place
b)will they be going in the same or opposite directions at the time they are in the same place?
c)at what value of t do they have the same velocity
d)during what time interval(s) are they moving in the same direction?

I need help with parts b and d

a)t^2+2t=-t^2+8t+80
2t^2+2t=8t+80
2t^2-6t-80=0
2(t^2-3t-40)=0
2(t-8)(t+5)=0
t=8 and t=-5, but we reject t=-5 because we cant have negative time

c) v1(t)=2t+2
v2(t)=-2t+8
2t+2=-2t+8
4t+2=8
4t=6
t=3/2

b) To determine whether the particles will be moving in the same or opposite directions at the time they are in the same place, we need to compare their velocities at that time. The velocity is the derivative of the position function with respect to time.

The velocity of particle 1 is given by v1(t) = 2t + 2.
The velocity of particle 2 is given by v2(t) = -2t + 8.

To find the time at which the particles are in the same place (as found in part a), we let t = 8 into the position functions and find that the particles meet at that time. Plugging this value of t into the velocity functions, we get:

v1(8) = 2(8) + 2 = 18
v2(8) = -2(8) + 8 = -8

Since the velocities have different signs (positive for particle 1 and negative for particle 2), the particles will be moving in opposite directions at the time they are in the same place.

d) To determine during what time interval(s) the particles are moving in the same direction, we need to compare their velocities throughout the given time range.

The velocity of particle 1 is given by v1(t) = 2t + 2.
The velocity of particle 2 is given by v2(t) = -2t + 8.

To find when they have the same velocity, we set v1(t) = v2(t) and solve for t:

2t + 2 = -2t + 8
4t = 6
t = 3/2

Therefore, the particles have the same velocity at t = 3/2.

To determine when they are moving in the same direction, we need to compare the signs of their velocities. Looking at the velocity functions, we can see that both velocities are positive when t > 3/2. Therefore, the particles are moving in the same direction for t > 3/2.