A surgical laser is cutting an incision in the eye according to the equation x^2-4y^2=9, y>0.
The incision rates vary to allow for curvature of the eye. When x=5, dx/dt=300 manometers per second. What is dy/dt at that moment?
A. 18.75 manometers per second
B. 93.75 manometers per second
C. 187.5 manometers per second
D. 960 manometers per second
x^2-4y^2=9,
2x dx/dt-8y dy/dt=0
when x=5, solve for y in the original equation, then put y in
2x dx/dt-8y dy/dt=0 and solve for dy/dt
To find dy/dt at the moment x = 5, we need to find the value of dy/dt by differentiating the given equation with respect to t. Here's how you can solve it step by step:
1. Start with the given equation: x^2 - 4y^2 = 9.
2. Differentiate both sides of the equation with respect to t. Since x and y are both functions of t, we have to use the chain rule for differentiation.
- Differentiating x^2 with respect to t gives 2x * dx/dt.
- Differentiating -4y^2 with respect to t gives -8y * dy/dt.
- Differentiating 9 with respect to t gives 0 because it's a constant.
So, we have: 2x * dx/dt - 8y * dy/dt = 0.
3. Now we need to substitute the given values: x = 5 and dx/dt = 300 into the equation.
- 2(5) * 300 - 8y * dy/dt = 0.
- 10 * 300 - 8y * dy/dt = 0.
- 3000 - 8y * dy/dt = 0.
4. Rearrange the equation to isolate dy/dt by moving the constant term to the other side.
-8y * dy/dt = -3000.
dy/dt = -3000 / (-8y).
Simplify dy/dt = 375 / y.
5. To find dy/dt at x = 5, we need to determine the value of y. To do so, substitute x = 5 into the original equation x^2 - 4y^2 = 9 and solve for y.
5^2 - 4y^2 = 9.
25 - 4y^2 = 9.
-4y^2 = -16.
y^2 = 4.
y = 2 because y > 0.
6. Substitute y = 2 into the equation dy/dt = 375 / y.
dy/dt = 375 / 2.
dy/dt = 187.5 manometers per second.
Therefore, the correct answer is C. 187.5 manometers per second.