Question

Two cars start moving from the same point. One travels south at 16 mi/h and the other travels west at 12 mi/h. At what rate is the distance between the cars increasing three hours later?

Answer 1

Answer:

20 miles per hour

Step-by-step explanation:

The distances traveled by each car are perpendicular, so we can find the distance between the cars using the Pythagoras' theorem between their distances traveled:

$d^2 = d_1^2 + d_2^2$

Where d is the distance between the cars, d1 is the distance traveled by the first car and d2 is the distance traveled by the second car.

The distance traveled is calculated by the speed times the time traveled, so we have:

$d^2 = (16t)^2 + (12t)^2$

$d^2 = 256t^2 + 144t^2$

$d^2 = 400t^2$

$d = 20t$

The rate that the distance is increasing can be found with the derivative of the distance in relation to the time:

$dd/dt = 20\ mph$

So the rate that the distance increases is always 20 miles per hour, and it's independent of the time.