Question

Suppose a car is traveling with variable velocity for one hour, and that
• At some point between 0 and 12 minutes, the car is moving with velocity 40 mph.
• At some point between 12 and 30 minutes, the car is moving with velocity −10 mph.
• At some point between 30 and 40 minutes, the car is moving with velocity 20 mph.
• At some point between 40 and 60 minutes, the car is moving with velocity 0 mph.
Use Riemann sums to approximate the total displacement of the car from its initial position.

Answer 1

At some point between 0 and 12 minutes, the car is moving with velocity 40 mph.

$S_1 = 40 \cdot (12 - 0) / 60 = 8$

At some point between 12 and 30 minutes, the car is moving with velocity −10 mph.

$S_2 = 10 \cdot (30 - 12) / 60 = 3$

At some point between 30 and 40 minutes, the car is moving with velocity 20 mph.

$S_3 = 20 \cdot (40 - 30) / 60 = 10/3 = 3 \cfrac{1}{3} = 3.333$

At some point between 40 and 60 minutes, the car is moving with velocity 0 mph.

$S_4 = 0 * (60 - 40) / 60 = 0$

And the total displacement of the car from its initial position, will be:

$S = S_1 + S_2 + S_3 + S_4 = 8 + 3 + 3 \cfrac{1}{3} = 14 \cfrac{1}{3} = 14.333$ miles

Answer: 14.333 miles