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To solve the problem, we need to differentiate the given equation with respect to \( t \). The equation is:

\[
x^2 + y^2 = 100
\]

Now, we will implicitly differentiate both sides of this equation with respect to \( t \):

\[
\frac{d}{dt}(x^2 + y^2) = \frac{d}{dt}(100)
\]

Using the chain rule:

\[
2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0
\]

Simplifying:

\[
x \frac{dx}{dt} + y \frac{dy}{dt} = 0
\]

This is the general equation we will use to solve for the required values in parts (a) and (b).

### Part (a):
We are given:
- \( x = 6 \)
- \( y = 8 \)
- \( \frac{dx}{dt} = 9 \)

Substitute these values into the equation \( x \frac{dx}{dt} + y \frac{dy}{dt} = 0 \):

\[
6(9) + 8 \frac{dy}{dt} = 0
\]

\[
54 + 8 \frac{dy}{dt} = 0
\]

Solve for \( \frac{dy}{dt} \):

\[
8 \frac{dy}{dt} = -54
\]

\[
\frac{dy}{dt} = -\frac{54}{8} = -6.75
\]

So, the value of \( \frac{dy}{dt} \) is \( -6.75 \).

### Part (b):
We are given:
- \( x = 8 \)
- \( y = 6 \)
- \( \frac{dy}{dt} = -2 \)

Substitute these values into the equation \( x \frac{dx}{dt} + y \frac{dy}{dt} = 0 \):

\[
8 \frac{dx}{dt} + 6(-2) = 0
\]

\[
8 \frac{dx}{dt} - 12 = 0
\]

Solve for \( \frac{dx}{dt} \):

\[
8 \frac{dx}{dt} = 12
\]

\[
\frac{dx}{dt} = \frac{12}{8} = 1.5
\]

So, the value of \( \frac{dx}{dt} \) is \( 1.5 \).

Let me know if you'd like further details or have any questions! Here are 5 additional related questions you might consider:

1. How does implicit differentiation apply when variables depend on time?
2. How does the chain rule work in multivariable functions?
3. What happens to the equation if we change the constant on the right-hand side to something else?
4. Can we derive a general formula for related rates in circular motion?
5. How would we solve the problem if the relationship between \( x \) and \( y \) were not as simple as a circle?

**Tip:** When working with related rates, always ensure that you substitute all known values after differentiating the entire equation.

Assume that x and y are both differentiable functions of t. Find the required values of dy/dt and dx/dt given the equation x^2 + y^2 = 100. Part (a) requires finding dy/dt given x = 6, y = 8, and dx/dt = 9. Part (b) requires finding dx/dt given x = 8, y = 6, and dy/dt = -2.

Learn how to solve a related rates problem involving a circle with the equation x^2 + y^2 = 100. Follow detailed steps to find dy/dt and dx/dt using implicit differentiation and the chain rule. This solution is suitable for students in Grades 10-12.

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