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.

Still not getting this! My teacher has me so confused.

right now i have 2L(dL/dt)=2(150)(300)sin(theta)(dtheta/dt)
I think that the angle changes at 7/150 but idk =[

See repost:

http://www.jiskha.com/display.cgi?id=1285961612

To find how fast the distance between the friends is changing, we need to find the rate of change of the distance with respect to time. Let's call the distance between the runner and their friend "d", and let's call the time "t". We'll need to apply the chain rule of differentiation.

First, let's set up a triangle to represent the situation. The distance between the friends is the hypotenuse of a right triangle, with the radius of the track as one of the legs. The other leg is the distance from the center of the track to the friend. Let's call the angle between the radius and the line connecting the friends θ.

Now, let's consider the rate of change of the distance between the friends. We can express this as dẋ, where ẋ represents the change in distance covered by the runner. This means that dẋ = 7 m/s, as the runner is sprinting at a constant speed of 7 m/s.

To find the rate of change of the distance between the friends, we need to find the derivative of the distance with respect to time (dd/dt). To do this, we can consider the relationship between the angle θ and the distance d.

Using the trigonometric relationship, we have d = √(r^2 + x^2), where r is the radius of the track and x is the distance from the center of the track to the friend. Since we know that x = 300 m, we can substitute this value into the equation.

d = √(150^2 + 300^2) = √(45000 + 90000) = √135000 = 366.96 m

Now, we'll differentiate both sides of the equation with respect to time (t), using the chain rule.

dd/dt = (1/2) * (45000 + 90000)^(-1/2) * (0 + 2 * 300 * (dx/dt))

Simplifying the equation, we get:

dd/dt = (1/2) * (45000 + 90000)^(-1/2) * 2 * 300 * dx/dt

dd/dt = (1/2) * (135000)^(-1/2) * 2 * 300 * 7

dd/dt = (1/2) * (135000)^(-1/2) * 2 * 300 * 7

dd/dt = (1/2) * (1/√135000) * 2 * 300 * 7

dd/dt = (1/√135000) * 300 * 7

Simplifying further, we get:

dd/dt ≈ 1.28

So, when the distance between the friends is 300 m, the distance is changing at a rate of approximately 1.28 m/s.

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