An athlete is running around a circular track of radius 50 meters at the rate of 5 meters per second. A spectator is 200 meters from the center of the track. How fast is the distance between the two changing when the runner is approaching the spectator and the distance between them is 200 meters?
I don't what equations to use.
To solve this problem, we can use the concept of related rates. The key is to find an equation that relates the variables involved in the problem: the distance between the runner and the spectator (let's call it D) and the time (t).
We're given that the athlete is running at a constant rate of 5 meters per second, so the rate at which the athlete is moving along the track is 5 meters per second (let's call it dr/dt).
Since the track is circular and the distance between the runner and the spectator is also changing, we need to consider the rate at which the distance between them is changing (let's call it dD/dt).
We can start by using the Pythagorean theorem to relate the distance between the runner and the center of the track (which is 50 meters) and the distance between the spectator and the center of the track (which is 200 meters):
D^2 = (50 + r)^2 + 200^2
Now, let's differentiate both sides of this equation with respect to time (t):
2D * dD/dt = 2(50 + r) * dr/dt
We're interested in finding dD/dt, so let's isolate it:
dD/dt = (50 + r) * dr/dt / D
Now we can plug in the values we know:
r = 200 meters (distance between the runner and the center of the track)
dr/dt = 5 meters per second (rate at which the runner is moving along the track)
D = 200 meters (distance between the runner and the spectator)
Plugging these values into the equation, we get:
dD/dt = (50 + 200) * 5 / 200
Simplifying this equation, we find:
dD/dt = 250 / 200 = 1.25 meters per second
So, when the runner is approaching the spectator and the distance between them is 200 meters, the distance between them is changing at a rate of 1.25 meters per second.
To solve this problem, we can use the concept of related rates. Let's denote the distance between the athlete and the spectator as "d" and the angle between the line connecting the athlete and the spectator and the line connecting the athlete and the center of the track as "θ".
First, let's find an equation relating the variables involved. We can use the fact that the athlete is running around a circular track. The distance the athlete travels along the track is equal to the circumference of the circle, which is 2π times the radius (50 meters).
Since the athlete is running at a rate of 5 meters per second, the rate at which the angle θ is changing can be determined by differentiating the equation with respect to time:
dθ/dt = 5 / (2π * 50)
Next, let's express the distance "d" between the athlete and the spectator in terms of θ using trigonometry. We can observe that the distance "d" is the hypotenuse of a right triangle, with the center of the track as the right-angle vertex and the line connecting the center and the athlete as the adjacent side. Therefore:
d = √(200² + 50² - 2 * 200 * 50 * cos(θ))
Now, let's differentiate this equation with respect to time (t) using the chain rule:
dd/dt = (d/dθ) * (dθ/dt)
We know dθ/dt, which we calculated earlier (5 / (2π * 50)), and we need to find dd/dt when d = 200.
Substituting the values in:
dd/dt = (d/dθ) * (dθ/dt)
= d(√(200² + 50² - 2 * 200 * 50 * cos(θ))) / dθ * 5 / (2π * 50)
Now, we can substitute the value of d (200) into this equation:
dd/dt = d(√(200² + 50² - 2 * 200 * 50 * cos(θ))) / dθ * 5 / (2π * 50)
= 200(√(200² + 50² - 2 * 200 * 50 * cos(θ))) / (5 / (2π * 50))
Simplifying further:
dd/dt = 200(√(40000 + 2500 - 200000 * cos(θ))) / (5 / (2π * 50))
= 400π(√(40250 - 40000 * cos(θ)))
Now, we can calculate the value of dd/dt when the angle θ = 0 degrees (or when the athlete is approaching the spectator and the distance between them is 200 meters) by substituting θ = 0 into the equation:
dd/dt = 400π(√(40250 - 40000 * cos(0)))
= 400π(√(40250 - 40000))
= 800π meters per second
Therefore, the distance between the athlete and the spectator is changing at a rate of 800π meters per second when they are 200 meters apart and the athlete is approaching the spectator.