If a point moves on the hyperbola x^2-4y^2=36 in such a way that x-coordinate increases at constant rate of 20 units per second. How fast is the y coordinated changing at point (10,4)?
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Differentiating implicitly with respect to t,
2 x dx/dt - 4 y dy/dt = 0
dy/dt = (1/2) (x/y) dx/dt
Plug in x=10, y=4 and dx/dt = 20 and crank out the answer.
dy/dt = 10 units per second
To find the rate at which the y-coordinate is changing at the point (10,4), we can use implicit differentiation.
Given the equation of the hyperbola: x^2 - 4y^2 = 36
Differentiate both sides of the equation with respect to time (t):
2x * dx/dt - 4y * dy/dt = 0
Now we can solve for dy/dt, which represents the rate at which the y-coordinate is changing:
dy/dt = (1/2) * (x/y) * dx/dt
Substituting x = 10, y = 4, and dx/dt = 20:
dy/dt = (1/2) * (10/4) * 20
Simplifying:
dy/dt = 5 * 5 * 20
dy/dt = 500 units per second
Therefore, the y-coordinate is changing at a rate of 500 units per second at the point (10,4) on the hyperbola.
To find the rate at which the y-coordinate is changing, we'll use implicit differentiation. We're given the equation of a hyperbola, x^2 - 4y^2 = 36, and we need to find how fast the y-coordinate is changing at the point (10,4) when the x-coordinate is increasing at a constant rate of 20 units per second.
First, let's differentiate both sides of the equation with respect to time (t):
d/dt (x^2 - 4y^2) = d/dt (36)
To differentiate x^2 with respect to t, we'll use the chain rule: d/dt (x^2) = 2x * dx/dt.
To differentiate -4y^2 with respect to t, we'll also use the chain rule: d/dt (-4y^2) = -8y * dy/dt.
Since the right side of the equation is a constant, its derivative is 0.
So, we have:
2x * dx/dt - 8y * dy/dt = 0
Now, we will solve this equation for dy/dt, which represents the rate at which the y-coordinate is changing. Let's isolate dy/dt:
-8y * dy/dt = -2x * dx/dt
Dividing both sides by -8y:
dy/dt = (1/2) * (x/y) * dx/dt
Now we can plug in the values we have: x = 10, y = 4, and dx/dt = 20:
dy/dt = (1/2) * (10/4) * 20
Simplifying:
dy/dt = (1/2) * (5/2) * 20
dy/dt = 0.5 * 2.5 * 20
dy/dt = 25
Therefore, at the point (10,4), the y-coordinate is changing at a rate of 25 units per second.