Question

A particle moves along a hyperbola xy=12. As it reaches the point (6,2), the y-coordinate is decreasing at a rate of 5 cm/s. How fast is the x-coordinate of the point changing at that instant? cm/s

Answer 1

To find the rate at which the x-coordinate is changing at the point (6,2) on the hyperbola xy=12, we use implicit differentiation and the given information. The x-coordinate is changing at a rate of 15 cm/s.

Step-by-step explanation:

To find the rate at which the x-coordinate is changing, we need to use implicit differentiation. We start with the equation of the hyperbola, xy = 12, and differentiate both sides with respect to time:

d(xy)/dt = d(12)/dt

Using the product rule and the chain rule, we get:

x(dy/dt) + y(dx/dt) = 0

Plugging in the given values, x = 6 and dy/dt = -5, we can solve for dx/dt:

6(-5) + 2(dx/dt) = 0

Simplifying, we find dx/dt = 15 cm/s.

Answer 2

The x-coordinate of the point is changing at a rate of 15 cm/s at that instant.

Step-by-step explanation:

To find the rate at which the x-coordinate of the point is changing when the y-coordinate is decreasing, we can differentiate the equation of the hyperbola xy = 12 with respect to time. Using the product rule, we get dx/dt * y + x * dy/dt = 0. Since we are given that dy/dt = -5 cm/s and we want to find dx/dt, we can substitute these values into the equation and solve for dx/dt.

At the point (6,2), we know that xy = 12, so substituting these values into the equation gives us 6 * 2 = 12. Differentiating both sides of this equation with respect to time, we get 6 * dy/dt + 2 * dx/dt = 0. Substituting the value of dy/dt = -5 cm/s, we can solve for dx/dt.

Plugging in the values, we have 6 * (-5 cm/s) + 2 * dx/dt = 0. Simplifying this equation, we get -30 cm/s + 2 * dx/dt = 0. Rearranging the equation, we find that 2 * dx/dt = 30 cm/s, and dividing both sides by 2, we get dx/dt = 15 cm/s. Therefore, the x-coordinate of the point is changing at a rate of 15 cm/s at that instant.