'Two straight roads meet at an angle of 60 degrees. A starts from the intersection and travels along one road at 40 km/hr. One hour later B starts from the intersection and travels along the other road at 50 km/hr. At what rate is the distance between them changing three hours after A starts?'

The answer the book gives is 43.1 km/hr.

da/dt = 40 and db/dt = 50. Letting t = 0 be the point where A is at the intersection, a = 0 + 40t and b = -50 + 50t.

The first method I tried was to convert the movement of a into x and y co-ordinates using the sine rule such that b is moving along the x axis and thus using pythagoras' theorem to find the formula for distance between the two points and differentiate it. Letting b = x, a

_{x}= 20t and a

_{y}= 20sqrt(3)t. I got dr/dt = 490t+350/sqrt(49t^2+70t+37). When t = 3 dr/dt = 455/sqrt(43) approximates 69.4 km/hr. No good.

The second method I tried was to use the cosine rule, drawing a and b as sides of a triangle changing with time, the distance between the two points c the third angle of the triangle opposing an angle of 60 degrees. Thus, r = sqrt(a^2+b^2-ab). This yielded r = 10sqrt(21t^2+15t+25), a curiously different equation from my last. Differentiating gave dr/dt = 210t+75/sqrt(21t^2+15t+25). When t = 3, dr/dt = 705/sqrt(259) approximates 43.8 km/hr, close but certainly not the right answer.

As you can see I am perplexed and baffled as to how two different methods are giving me two different equations, none of which is right. What am I doing wrong in each approach?

EDIT: I got my second method to give the correct answer by mastering the daunting task of adding and subtracting numbers correctly. I am still morbidly curious as to what is wrong with my first approach, however.