Calculus problem

Question

Two cars, initially are 130 miles apart, start off at the same time. Car A is to the west of Car B and starts traveling east (towards Car B) at 25 miles per hour while at the same time Car B starts traveling south at 10 miles per hour. After 4 hours of traveling at what rate is the distance between the two cars charging? Is it increasing or decreasing?

Answer 1

The representative figure is a right triangle with right angle at Origin, initial x value = -130, initial y value = 0 (endpoints of legs of triangle).

 

dx/dt is given as positive 25 (because the negative x coordinate is increasing due to Eastward movement); 

dy/dt is stated as -10 mph because car B is heading south.

 

The hypotenuse has length z and x² + y² = z² and from this we get

z(dz/dt) = x(dx/dt) + y(dy/dt) after doing implicit differentiation & dividing each term by 2 to simplify.

 

At the instant in question, x= -30 (because 4 hrs of travel at 25 mph increased x by 100),

y = -40, and z = 50 miles (note the "3-4-5" triangle).

 

dz/dt = (1/50)[(-30)(25) + (-40)((-10)] = -350 / 50 = -7 mi/hr; the distance between cars is decreasing @ 7 mph.