To solve for dx/dt when y = 6, we start with the given equation:
x² + y² = 100
First, we differentiate both sides of the equation with respect to t:
2x(dx/dt) + 2y(dy/dt) = 0
We can simplify this equation by dividing everything by 2:
x(dx/dt) + y(dy/dt) = 0
Now, we can rearrange this to solve for dx/dt:
dx/dt = – (y(dy/dt))/x
Next, we need to find the value of x when y = 6. We plug y = 6 into the original equation:
x² + 6² = 100
x² + 36 = 100
x² = 100 – 36
x² = 64
x = 8 (since we are generally interested in positive values)
Now we can substitute x = 8, y = 6, and dy/dt = 4 into our equation for dx/dt:
dx/dt = – (6 * 4) / 8
dx/dt = – 24 / 8
dx/dt = -3
Therefore, the rate of change of x with respect to t when y = 6 is -3.