related rates question

Question

Two cars start moving from the same point. One travels south at 32 mi/h and the other travels west at 24 mi/h. At what rate is the distance between the cars increasing four hours later?

Answer 1

draw your figure

This is a right triangle where x²+y²=d² where x is the distance traveled by the car going West and y is the distance traveled by the car going south

We have two rates dx/dt=24mi/h and dy/dt=32mi/h

We want to find the distance at 4 hours using this information at 4 hours x=(24)(4)=96mi y=(32)(4)=128mi, d=160mi.

We take the derivative of the equation to figure out the relationship between the rates

x²+y²=d²

2x(dx/dt)+2y(dy/dt)=2d(dd/dt)

2(96)(24)+2(128)(32)=2(160)(dd/dt)

Solve for dd/dt

dd/dt=40mi/h