When does object A overtake object B?

Question

Two objects at t=0t = 0t=0 start moving along a straight path. Object A starts at 8 miles per hour.

1. The velocity of object A changes according to the function Va(t)= (t+8)/(e^t) for t>0.
2. The velocity of object B changes according to the function Vb(t)=(2t)/(1+t^4) for t>0.

Object B starts with a 3 mile head start on object A. When, if ever, does object A overtake object B? Use integration to find the answer.

I need help!

Thank you!

Answer 1

∫^t₀V_a(x)dx = ∫^t₀V_b(x)dx

D_a(t)|^t₀ = D_b(t)|₀^t

D_a(0) = 0

D_b(0) = 3

Then solve for t. You might need to use Newton's method to solve for t, or plot the integrated functions to see where D_a(t) and D_b(t) intersect.

Answer 2

If you integrate each velocity function from 0 to t and add the 3 mile head start, you should get

9-te^-t-9e^-t = 3+arctan t²

^{Then use a graphing calculator to get the result.}

^{I got t=.5}

^{P.S. I could integrate the first one myself, but I had to look up the 2nd one!}