After t hours on a particular day on the railways of the Isl

After t hours on a particular day on the railways of the Island of Sodor, Thomas the Tank Engine is s(t)= -.03t^3+5t^2-2t miles east of Knapford Station (for 0t16)

1. Where is Thomas (both distance and direction from the starting point at Knapford Station) after 8 hours?

2. What is the velocity of Thomas after 8 hours?

3. What is the acceleration of Thomas after 8 hours?

4. Is the Thomas speeding up or slowing down after 8 hours? How can you tell?

Solution

given s(t)= -.03t^3+5t^2-2t

1. after t=8

s(8) = -0.03(8³)+5(8²)-2(8)

= 288.64 miles

2. velocity v(t) = ds(t)/dt = d/dt[ -.03t^3+5t^2-2t ] = - 0.09t² + 10t - 2

at t = 8 hours, v(8) = 72.24 miles/hour

3. accelaration a(t) = d/dt[v(t) ] = d//dt [  - 0.09t² + 10t - 2 ] = - 0.18t + 10

at t=8hours , a(8) = 8.56 miles/hours²

4 . speed = distance / time =  [-.03t^3+5t^2-2t ]/t =  -.03t^2+5t-2

at t= 8 hours , speed = 36.08 miles/hours