Question

Two cars start moving from the same point. One travels south at 60 min/h and the other travels west at 25 min/h. At what rate is the distance between the cars increasing three hours later?

Answer 1

Distance is increasing at 65 mi/hr after 3 hours.

Explanation:

They are travelling perpendicularly.

One travels south at 60 mi/h and the other travels west at 25 mi/h.

We need to find at what rate is the distance between the cars increasing three hours later.

After 3 hour distance traveled by car 1 = 60 x 3 = 180 miles

After 3 hour distance traveled by car 2 = 25 x 3 = 75 miles

`\texttt{Distance between cars after 3 hours = }√(180^2+75^2)=195miles`

Let the distance be s, distance by car be s₁, and distance by car 2 be s₂

We have

s² = s₁²+s₂²

Differentiating

`2s(ds)/(dt)=2s_1(ds_1)/(dt)+2s_2(ds_2)/(dt)\\\\s(ds)/(dt)=s_1(ds_1)/(dt)+s_2(ds_2)/(dt)\\\\s=195mi\\\\s_1=180mi\\\\s_2=75mi\\\\(ds_1)/(dt)=60mi/hr\\\\(ds_2)/(dt)=25mi/hr`

Substituting

`s(ds)/(dt)=s_1(ds_1)/(dt)+s_2(ds_2)/(dt)\\\\195* (ds)/(dt)=180* 60+75* 25\\\\(ds)/(dt)=65mi/hr`

Distance is increasing at 65 mi/hr after 3 hours.