Question

A jogger runs around a circular track of radius 55 ft. Let (x,y) be her coordinates, where the origin is the center of the track. When the jogger's coordinates are (33, 44), her x-coordinate is changing at a rate of 15 ft/s. Find dy/dt.

Answer 1

Using related rates and the Pythagorean theorem, we can find that dy/dt, the rate of change of the y-coordinate, is 0 ft/s.

Step-by-step explanation:

To find dy/dt, we need to determine the rate of change of the y-coordinate of the jogger. Since the jogger is running on a circular track, we can use the concept of related rates to solve this problem.

Let's assume that the jogger completes a full lap around the track in a time interval of Δt. During this time interval, the x-coordinate of the jogger changes by Δx, the y-coordinate changes by Δy, and the distance traveled along the track is Δs.

Since the jogger is running at a constant speed, the distance Δs is equal to the distance traveled in a straight line, which is the hypotenuse of a right triangle with legs Δx and Δy. Using the Pythagorean theorem, we have:

Δs^2 = Δx^2 + Δy^2

Taking the derivative with respect to time, we have:

2Δs(dΔs/dt) = 2Δx(dΔx/dt) + 2Δy(dΔy/dt)

Substituting the given values, Δx is 15 ft/s, Δy is 0 (since the y-coordinate is not changing), and Δs is the distance around the circular track, which is equal to the circumference of the circle:

2π(55ft) = 2(15ft)(dΔx/dt) + 2(0)(dΔy/dt)

Simplifying, we have:

dΔx/dt = π(55ft)/15s = 11π/3 ft/s

Therefore, dy/dt = dΔy/dt = 0 ft/s, since the y-coordinate is not changing.

Answer 2

11.25 ft/sec

Step-by-step explanation:

Given that a jogger runs around a circular track of radius 55 ft.

. Let (x,y) be her coordinates, where the origin is the center of the track.

Then we know x,y satisfies the equation of the circle as

`x^2+y^2 = 55^2`

Let us differentiate this implicit function with respect to t

`2x (dx)/(dt) +2y (dy)/(dt) =0`

At this point, dx/dt 15 and x=33. y =44

Substitute to have

`2(33)(15) +2(44) (dy)/(dt)\\=0\\(dy)/(dt)=11.25`

dy/dt = 11.25 ft /s