derrivative calculus

Question

Two straight roads diverge from an intersection at an angle of 60°. Two cars leave the intersection at the same time, the first traveling down one road at 44 mi/h and the second traveling down the other road at 60 mi/h. How fast (in mi/h) is the distance between the cars changing after half an hour? [Hint: Use the Law of Cosines.] (Round your answer to one decimal place.)

Answer 1

Hello Hunter,

For your question. You will definitely need to refer to the Law of Cosines.

Let's suppose we have a triangle of some kind such that one vertice is the intersection. I will call it vertice A so it's easy to reference. Now suppose we have the first car going down direction 'c' (side c of the triangle coming from vertice A) and the second car going down direction 'b' (side b of the triangle coming from vertice A) where dc/dt = 44 mi/h (the speed of the first car) and db/dt = 60 mi/h (the speed of the second car).

Our unknown now is the side 'a' which we are trying to find da/dt since as the cars continue to diverge away from one another, side 'a' continues to grow.

For the law of cosine:

a^2 = b^2 + c^2 - 2bc*cos(A)

The 'A' in our case from the vertice A will be 60 degrees which was given to us.

Recall that cos(A = 60 degrees) = 1/2

Now, our relation equation will be:

a^2 = b^2 + c^2 - 2bc*(1/2) = b^2 + c^2 - bc (simplified)

We are still missing the length of side 'b', side 'c', and side 'a' before we take our derivative. We can solve for side 'b' and side 'c' with our given information and solving for 'b' and 'c' will also give us side 'a'.

We use our given info: t = half an hour (30 min)

Since our units for db/dt and dc/dt are in mi/h, it's simply asking us how much distance the cars have traveled after 30 minutes.

So, for side 'b', db(30)/dt = 30 miles and for side 'c', dc(30)/dt = 22 miles.

Now solving for side 'a'. Refer back to our relation equation:

a^2 = b^2 + c^2 - bc

a^2 = (30)^2 + (22)^2 - (30)(22) = 724

a = sqrt(724)

Now, we can actually take our derivative for our relation equation:

d/dt(a^2 = b^2 + c^2 - bc)

2a*(da/dt) = 2b*(db/dt) + 2c*(dc/dt) - (product rule on 'bc')

I will leave the rest up to you as it's just finishing the product rule and you have every single component needed to plug in to solve for da/dt.

Hope this helps!