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.

Step-by-step 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

$\texttt{Distance between cars after 3 hours = }\sqrt{180^2+75^2}=195miles$

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

We have

                        s² = s₁²+s₂²

Differentiating

                      $2s\frac{ds}{dt}=2s_1\frac{ds_1}{dt}+2s_2\frac{ds_2}{dt}\\\\s\frac{ds}{dt}=s_1\frac{ds_1}{dt}+s_2\frac{ds_2}{dt}\\\\s=195mi\\\\s_1=180mi\\\\s_2=75mi\\\\\frac{ds_1}{dt}=60mi/hr\\\\\frac{ds_2}{dt}=25mi/hr$

Substituting

            $s\frac{ds}{dt}=s_1\frac{ds_1}{dt}+s_2\frac{ds_2}{dt}\\\\195\times \frac{ds}{dt}=180\times 60+75\times 25\\\\\frac{ds}{dt}=65mi/hr$

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