please help with this calculus

Question

A car travels along a straight road for 30 seconds starting at time t = 0. Its acceleration in ft/sec² is given by the linear graph below for the time interval [0, 30]. At t = 0, the velocity of the car is 0 and its position is 10.

What is the position of the car when t = 20? You must show your work answer to 3 decimal places.

Answer 1

Think Maclaurin.

f(t) = f(0)+f'0)t + f''(0)t^2/2! + f'''0)t^3/3!, as f''''(t) = 0 for all t

f(0) = 10

f'(0) = 0

f''(x) = 10 - t, so f''(0) = 10 and f'''(t) = -1 for all t

f(t) = 10+ 0t + 10t^2/2! + -1t^3/3!

f(t) = 10 + 5t^2 -t^3/6

f(20) = 10 + 5 * 20^2 - 20^3/6 = 10 + 2000 - 8000/6 = 676 2/3