a particle is moving counterclockwise around the circular path whose equation is x^2+y^2=100 of the abscissa of the particle is increasing by 72in/sec at a point where x=8 find the rate of changeof the cordinate(y) at this point
the rate of ordinate change is dy/dt.
x^2 + y^2 = 100
2x dx/dt + 2y dy/dt = 0
when x=8, y=6, so
2(8)(72) + 2(6) dy/dt = 0
12 dy/dt = -1152
dy/dt = -96 in/s
To find the rate of change of the y-coordinate, we can differentiate the equation of the circular path with respect to x.
Differentiating both sides of the equation x^2 + y^2 = 100 with respect to x, we get:
2x + 2y(dy/dx) = 0
Rearranging the equation, we have:
dy/dx = -2x / 2y
Now, we can substitute the given value of x = 8 into the equation to find the rate of change of the y-coordinate at this point.
dy/dx = -2(8) / 2y = -16 / 2y = -8/y
To calculate the actual rate of change, we need to know the value of y at this point. Unfortunately, we don't have that information.
To find the rate of change of the y-coordinate at the point where x = 8, we need to differentiate the equation of the circular path with respect to time.
Given equation: x^2 + y^2 = 100
Differentiating both sides with respect to time (t), we get:
(2x)(dx/dt) + (2y)(dy/dt) = 0
Since the particle is moving counterclockwise around the circular path, the x-coordinate is increasing at a rate of 72 in/sec. Thus, dx/dt = 72.
Substituting the given values:
(2 * 8)(72) + (2y)(dy/dt) = 0
Simplifying the equation:
144 + (2y)(dy/dt) = 0
Now we can solve for the rate of change of the y-coordinate (dy/dt):
(2y)(dy/dt) = -144
dy/dt = -144 / (2y)
To find the value of y, we can substitute x = 8 into the equation of the circular path:
8^2 + y^2 = 100
64 + y^2 = 100
y^2 = 100 - 64
y^2 = 36
y = ±√36
y = ±6
Since the particle is moving counterclockwise, the y-coordinate is increasing. So we take the positive value:
y = 6
Substituting the values of y and solving for dy/dt:
dy/dt = -144 / (2 * 6)
dy/dt = -144 / 12
dy/dt = -12
Therefore, the rate of change of the y-coordinate at the point (8, 6) is -12 units per second.