Julian is jogging around a circular track of radius 50 m. In a coordinate system with its origin at the center of the track, Julian's x-coordinate is changing at a rate of -1.25 m/s when his coordinates are (40, 30). Find dy/dt at this moment.
user
Sure, I'd be happy to help with this problem. In calculus, the pace of change of a function f(x,y) with respect to y is denoted by the partial derivative ∂f/∂y.

In your case, the function being considered is y, which is one of the variables along with x, and both these variables change with time. Therefore, it is a multivariate calculus problem.

Assuming that the y-coordinate represents Julian's height measured upward from the ground instead of the y-axis in a 2D coordinate system. In that case, the partial derivative denotes the rate at which y changes with respect to t in terms of x, y = f(x,t).

The problem is actually about finding the velocity of Julian in the direction of the y-axis (i.e., the direction of the track). The speed can be found by using Pythagoras' theorem (v = sqrt(dx/dt2 + dy/dt2) ), but it is impossible to determine without knowing how y changes. Of course, the problem is improperly formulated.

The correct problem would be:

"If Julian is jogging along a straight track of radius 50 meters, what is his velocity, applying the headwind or tailwind in the direction perpendicular to his track?"

Assume that the velocity's magnitude, v, remains "2d" (written in meters per second, m/s) and assume that the air pressure on the normal will return to 1 atmosphere.

This prize is dependent on how rapid Julian travels, so this calculation could also be interpreted as a calculation of the average velocity, the initial rate of change (determined by dx/dt= 1.25 and dy/dt = 2d, and v2= dx/dt2 + dy/dt2).

v2 = vx2+ vy2 = (1.25)^2 + (2d)^2 = 4.0625 + 4d2 ⇔ v = v√(1 + (2d/1.25)2), which means that v = 2.616 dm/s only for significant changes when d is near 0.

That said, it's not apparent how exactly it's supposed to be related to the problem. But I am assuming that since the problem involves circle jogging at a constant distance or a straight route and a changing thrust, you need to account for the force and speed using the force formula F = Δpv (providing pressure multiplied with pace times a directed 1-meter structure holding force). The PR is believed to be d, and by doing so, we can get the results we require."
assistant
Julian is jogging around a circular track of radius 50 m. In a coordinate system with its origin at the center of the track, Julian's x-coordinate is changing at a rate of -1.25 m/s when his coordinates are (40, 30). Find dy/dt at this moment.
user
We are given that Julian is jogging around a circular track of radius 50 m, and his x-coordinate is changing at a rate of -1.25 m/s when his coordinates are (40, 30). We want to find the rate of change of his y-coordinate (dy/dt) at this moment.

First, let's recall the equation for a circle centered at the origin:

x^2 + y^2 = r^2

Differentiating both sides with respect to time (t), we get:

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

Now, we can plug in the given values:

x = 40 m
y = 30 m
dx/dt = -1.25 m/s

So,

2(40)(-1.25) + 2(30)(dy/dt) = 0

Simplify and solve for dy/dt:

-100 + 60(dy/dt) = 0
60(dy/dt) = 100
dy/dt = 100 / 60
dy/dt ≈ 1.67 m/s

So, Julian's y-coordinate is changing at a rate of approximately 1.67 m/s at this moment.
assistant