need a little help

Question

Two straight roads diverge at an angle of 40°. Two cars leave the intersection at 2:00 P.M., one traveling at 38 mi/h and the other at 60 mi/h. How far apart are the cars at 2:30 P.M.? (Round your answer to two decimal places.)

 mi

Answer 1

One way to do this problem is to set up an x,y coordinate system in which the first road is along the x axis and the second road is in the first quadrant making the 40 deg angle with the x axis.

The the position of the first car is given by p1 = (38 t, 0) and the position of the second car is given by

p2 = ( 60 cos(40) t, 60 sin(40) t )

The distance formula then gives the distance between the cars as:

d = sqrt[ (38 t - 60 cos(40) t )^2 + ( 60 sin(40) t )^2 ]

Evaluating this for t = 1/2 gives: 19.69 miles

Answer 2

With the information given, we have to use the Law of Cosines to solve this one. Let the 40^o angle be A. The other two angles are B and C. (Uppercase indicates angle measures; lowercase, side lengths)

We can calculate two side lengths with the speeds and times given. 60 mph x 0.5 hr = 30 miles, let that be side b. 38 mph x 0.5 hr = 19 miles. Let that be side c.

Law of Cosines: a²= b²+c²-2bcCosA

Now plug in everything we know a²= 30²+ 19²-2(30)(19)Cos(40)

The rest is algebra, so you can take it from here.

Best wishes!