Question

3. Brust is riding his bicycle north away from an intersection at a rate of 15 miles per hour. Sully is driving his car towards the intersection from the west at a rate of 30 miles per hour. If Brust is 0.4 miles from the intersection, and Sully is 1 mile from the intersection, at what rate is the distance between the two of them increasing or decreasing?

Answer 1

The graph shows the situation of Brust and Sully. The distance between them is d

If x is the distance from Sully to the intersection and y is the distance from Brust to the intersection, the distance d is

$d=\sqrt[]{x^2+y^2}$

The rate of change of d in time is computed by taking the derivative:

$d^{\prime}=\frac{xx^{\prime}+yy^{\prime}\text{ }}{\sqrt[]{x^2+y^2}}$

We have the following parameters:

x=1, y=0.4, x'=-30, y'=15

Substituting:

$d^{\prime}=\frac{(1)(-30)+(0.4)(15)\text{ }}{\sqrt[]{1^2+0.4^2}}$

d' = -22.3 miles per hour

Since d' is negative, the distance is decreasing