Question

.
A car A is travelling from west at 50 km/hr. and car B is
travelling towards north at 60 km/hr. 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
kilometers and car B is 0.4 kilometers from the intersection?​

Answer 1

Answer= 7.80 mph . Let x =distance travelled by A and y=distance travelled by B where A traveling at western direction and y at northern direction forming a right triangle with sides x, y, and z as the hypotenuse.

Using Pythagorean’s Theorem:

x^2 + y^2 = z^2

Using derivatives:

2xdx/dt + 2ydy/dt= 2zdz/dt; where dx/dt=50 and dy/dt=60

xdx/dt+ydy/dt=zdz/dt

z=[xdx/dt+ydy/dt]/dz/dt

At the instant when x=0.3 mile, y=0.4 mile, determine the value of z by Pythagorean’s theorem:

(0.3)^2+(0.4)^2=z^2

.09+0.16=z^2

0.25=z^2

z=0.5 mile

substitute z=0.5

0.5=[0.3(50)+0.4(60)]/dz/dt

dz/dt=(15+24)/5= 39/5= 7.80 mph, rate of approaching each other. This is also the rate of change of distance z with respect to time.
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