why the direction to the north is negative?

Question

Car A is traveling west at 50 mi/h and car B is traveling north at 60 mi/h. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection?

let x= car A's way and y= car B's way.

In this problem, when I see the solution of it, I can see that dx/dt= -50mi/h and dy/dt= -60 mi/h because both x and y are decreasing. But I can't understand why the direction to the north means decreasing. If the north way is decreasing, then what if heading to the south way? Does the south way indicate increasing?

Answer 1

Car A moves west at 50 mph x'=50

Car B moves north at 60 mph y'=60

they will soon collide at an intersection, if they keep moving at the same speed

A is .3 miles from the intersection x=.3

B is .4 miles from the intersection y=.4

they are d miles apart, d=sqr(.3^2+.4^2) = sqr(.09+.16) = sqr.25 = .5 miles apart

d= .5

how fast are they approaching each other?

that's d', change in d with respect to time

using the Pythagorean theorem

d^2 = x^2 + y^2

take derivatives with respect to time

2dd' = 2xx' +2yy'

divide by 2

dd' = xx' +yy'

d' = (xx' +yy')/d, plug in the values x=.3, y=.4, x'=50, y'=60, d=.5

= (.3(50)+.4(60))/.5

= .6(50) +.8(60)

= 30+ 48

= 78 mph

they're getting closer to each other at the rate of 78 mph

were they headed directly at each other, they'd be getting closer at the rate of 50+60=110 mph

if headed in totally opposite direction, they'd never meet and be getting "closer" at the negative rate = -110 mph