In the problem, x and y are differentiable functions of t. Find dx/dt when x = 3, y = 4, and dy/dt= 2.
x^2 + y2^2= 25
To find dx/dt, we need to differentiate the equation x^2 + y^2 = 25 with respect to t.
1. Start by differentiating both sides of the equation using the chain rule.
On the left side, the derivative of x^2 with respect to t is given by 2x dx/dt.
On the right side, the derivative of 25 with respect to t is 0 since it's a constant.
So, the equation becomes: 2x dx/dt + 2y dy/dt = 0.
2. Now, substitute the given values into the equation.
When x = 3, y = 4, and dy/dt = 2, we have:
2(3) dx/dt + 2(4)(2) = 0.
Simplifying this equation gives:
6 dx/dt + 16 = 0.
3. Solve for dx/dt by isolating the dx/dt term.
Subtracting 16 from both sides of the equation, we get:
6 dx/dt = -16.
Dividing both sides of the equation by 6, we find:
dx/dt = -16/6 = -8/3.
Therefore, the value of dx/dt when x = 3, y = 4, and dy/dt = 2 is -8/3.