A runner sprints around a circular track of radius 150 m at a constant speed of 7 m/s. The runner's friend is standing at a distance 300 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 300 m? (Round to 2 decimal places.)

I get how to do related rates. But I have no idea where to start with this question. I know we have to use the formula c^2= a^2 + b^2 - 2(ab)cosC but that's it

your kind of really bad at this

To solve this problem using related rates, we can think of the runner and their friend as moving along arcs of circles on the track. We need to find the rate at which the distance between them changes, which is the derivative of the distance with respect to time.

Let's denote the distance between the runner and their friend as D, and the angle as θ. We are given that D = 300 m, which is constant, but we want to find dD/dt, the rate at which D changes with respect to time.

To start, we need to find an equation relating D and θ. Here's how we can do that:

1. Draw a diagram to visualize the problem. You have a circle representing the track, with a center and radius given. Mark the points where the runner and their friend are initially located. Label the distance between them as D.

2. Note that D is the hypotenuse of a right triangle. The base of the triangle is the radius of the track (150 m) and the height is the distance of the friend from the center of the track (also 300 m).

3. Use the Pythagorean theorem to relate D, the base, and the height of the triangle:

D^2 = (radius)^2 + (friend's distance from the center)^2

D^2 = 150^2 + 300^2

D^2 = 22500 + 90000

D^2 = 112500

D = √112500 = 335.41 (rounded to two decimal places)

So we have our equation relating D and the other given values.

Now, we need to find the derivative of D with respect to time (dD/dt). Since D is constant in this problem, it doesn't change with time, so dD/dt = 0.

Therefore, the rate at which the distance between the runner and their friend is changing is 0 m/s.

It's important to note that even though the runner is moving along the circular track, their distance from their friend remains constant because their friend is standing in the center of the track.