Calculus Question

Question

The velocity of a car (in miles per hour) is given by

v(t)=20t−10t^2

where t is in hours. Write a definite integral for the distance the car travels during the first 2 hours. Use your graphing calculator to find this distance. 

Answer 1

v(t) = -10t^2 + 20t

s(t) = the integral of v(t) = -(10/3)t^3 + 10t

s(2) = -(10/3)2^3 + 10(2) = -80/3 + 20 = -26.7+20 = -6.7 miles

max distance is when v'= 0 when -20t + 20 = 0 or t = 1 second

s(0) = 0

s(1) = -10/3 + 10 = 20/3 = 6 2/3 miles

s(2) = -6 2/3 miles

It reaches maximum distance at one second with 6 2/3 miles, then it slows down and goes backwards, At 2 seconds it's gone backwards 13 1/3 miles from its maximum distance.