A point is moving along the curve xy=12. When the point is at (4,3), the x-coordinate decreases at the rate of 2cm/sec. How fast is the y-coordinate changing at that point?
xy = 12
y dx/dt + x dy/dt = 0
Now just plug in your numbers and solve for dy/dt
To find how fast the y-coordinate is changing at the point (4, 3), we need to find the rate of change of y with respect to time (dy/dt) when the x-coordinate is changing at a certain rate.
We first differentiate the given equation xy = 12 with respect to t:
d(xy)/dt = d(12)/dt
Using the product rule of differentiation, we have:
x(dy/dt) + y(dx/dt) = 0
At the given point (4, 3), we can substitute the values:
4(dy/dt) + 3(-2) = 0
Simplifying the equation:
4(dy/dt) - 6 = 0
4(dy/dt) = 6
(dy/dt) = 6/4
(dy/dt) = 3/2
Therefore, the y-coordinate is changing at a rate of 3/2 cm/sec at the point (4,3).
To find how fast the y-coordinate is changing at a given point, we need to use the chain rule of differentiation. The curve given by xy = 12 can be rewritten as y = 12/x.
Let's denote the x-coordinate as x(t), where t represents time. We are given that dx/dt = -2 cm/sec, meaning that the x-coordinate is decreasing. We need to find dy/dt, the rate at which the y-coordinate is changing.
Using the chain rule, we have:
dy/dt = dy/dx * dx/dt
To find dy/dx, we differentiate y = 12/x with respect to x:
dy/dx = -12/x^2
Now we know that dx/dt = -2, and we can substitute it into the equation:
dy/dt = (-12/x^2) * (-2)
Plugging in the x-coordinate at the given point (4,3) into the equation:
dy/dt = (-12/4^2) * (-2) = (-12/16) * (-2) = 3/4 * 2 = 6/4 = 3/2 cm/sec.
Therefore, the y-coordinate is changing at a rate of 3/2 cm/sec at the point (4,3).