Two cars begin 500 miles apart.  Car A is into the west of Car B and begin driving to the east (that means towards Car B) at 35 mph & at the similar time Car B begin driving south at 50 mph.  After 3 hrs of driving at what rate is the distance among the two cars changing?  Is it raising or falling?

Solution : The first thing to do at this time is to get sketch a figure illustrating the situation.

In this figure y show the distance driven through Car B and x represents the distance separating

Car A from Car B's first position & z represents the distance separating the two cars.  After 3 hours driving time along with have the given values of x & y.

x = 500 - 35 (3) = 395

y = 50 (3) = 150

We can utilizes the Pythagorean theorem to determine z at this time as follows,

z ²  = 395² + 150²  = 178525   ⇒  z =   √178525 = 422.5222

Now, to give answer this question we will have to determine z′ given that x′ = -35  and y′ = 50 .  Remember that a rate is negative if the quantity is decreasing and positive if the quantity is increasing.

We can again use the Pythagorean theorem here.  Firstly, write it down and the remember that x, y, & z are all changing  along with time and thus differentiate the equation by using Implicit Differentiation.

              z ²  = x² + y ² ⇒ 2 zz′ = 2 xx′ + 2 yy′

At last, all we have to do is cancel a two from everything, plug in for the known quantities and solve out for z′ .

z′ = (422.5222) = (395) ( -35) + (150) (50)    ⇒ z′ = -6325 /422.5222= -14.9696

Thus, after three hours the distance among them is decreasing at rate of 14.9696 mph.