A jogger runs around a circular track of radius 65 ft. Let (x,y) be her coordinates, where the origin is the center of the track. When the jogger's coordinates are (39, 52), her x -coordinate is changing at a rate of 15ft/s. Find (dy/dx)
I think you want to find dy/dt
or else we don't need the fact that
dx/dt = 15 ft/s
x^2 + y^2 = 65^2
2x dx/dt + 2y dy/dx= 0
dy/dx = (-2x dx/dt)/(2y) = (-x dx/dt)/y
so for the given data
dy/dt = -39(15)/52 = -45/4 or -11.25 ft/s
Sure thing! I'll help you find (dy/dx), but let me put on my running shoes first. *Puts on clown shoes instead*
Now, we have a jogger running around a circular track with a radius of 65 ft. And at the specific coordinates (39, 52), the x-coordinate is changing at a rate of 15 ft/s.
To find (dy/dx), we'll need to use the relationship between x and y on the circle. This relationship is given by the equation of a circle: x^2 + y^2 = r^2, where r is the radius of the circle.
Since the jogger's coordinates are (39, 52), we can substitute these values into the equation: 39^2 + 52^2 = 65^2.
Now, let's differentiate both sides of the equation with respect to time t (because we want to find rates of change):
2x(dx/dt) + 2y(dy/dt) = 0
Since we're interested in finding (dy/dx), we can rearrange this equation to isolate it:
(dy/dt) = -x(dx/dt) / y
Substituting the given values, we have:
(dy/dt) = -39(15) / 52
(dy/dt) = -585 / 52
Hence, (dy/dx) is approximately equal to -11.25.
To find (dy/dx), we need to first express y in terms of x.
Since the origin is the center of the track, we can see that the jogger is always on a circle with a radius of 65 ft. This means that the distance from the origin to the jogger is always 65 ft.
Using the Pythagorean theorem, we can express y in terms of x:
y^2 + x^2 = 65^2
Taking the derivative with respect to x, we get:
2y(dy/dx) + 2x = 0
Now we can substitute the given values:
x = 39
y = 52
Plugging these values into the equation, we have:
2(52)(dy/dx) + 2(39) = 0
Simplifying further, we get:
104(dy/dx) + 78 = 0
Now we can solve for dy/dx:
104(dy/dx) = -78
dy/dx = -78/104
Simplifying, we get:
dy/dx = -3/4
Therefore, the rate at which the y-coordinate is changing when the jogger's x-coordinate is changing at a rate of 15 ft/s is -3/4 ft/s.