a point is moving on the graph of xy=30. when the point is at (6,5), its x-coordinate is increasing by 6 units per second. how fast is the y-coordinate changing at the moment
xy=constant
y dx/dt+ x dy/dt=0
you know x,y, and dx/dt solve for dy/dt
To find how fast the y-coordinate is changing at the moment, we can use the implicit differentiation method.
The given equation is xy = 30. Taking the derivative of both sides with respect to time (t), we get:
x(dy/dt) + y(dx/dt) = 0
Since we are given that the point is at (6,5) and its x-coordinate is increasing by 6 units per second (dx/dt = 6), we can substitute these values into the equation:
(6)(dy/dt) + (5)(6) = 0
Simplifying the equation:
6(dy/dt) + 30 = 0
Now, we can isolate dy/dt by subtracting 30 from both sides and dividing by 6:
6(dy/dt) = -30
dy/dt = -30/6
dy/dt = -5
Therefore, the y-coordinate is changing at a rate of -5 units per second at the given moment.