To find how fast the temperature is rising along the bug's path, we need to use the chain rule for multivariable functions. Here's how we approach the problem:

Given Information:

• The temperature at a point $(x, y)$ is $T(x, y)$, where $x = 1 + t$ and $y = 8 + \frac{1}{8}t$.
• The rates of change of temperature with respect to $x$ and $y$ are given as $T_x(3, 9) = 5$ and $T_y(3, 9) = 9$.
• We want to find the rate at which the temperature is changing with respect to time, $\frac{dT}{dt}$, when $t = 8$.

Step 1: Use the Chain Rule

We want to compute $\frac{dT}{dt}$ using the chain rule:

$\frac{dT}{dt} = \frac{\partial T}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial T}{\partial y} \cdot \frac{dy}{dt}$

Step 2: Compute $\frac{dx}{dt}$ and $\frac{dy}{dt}$

From the given position functions:

• $x = 1 + t$ implies $\frac{dx}{dt} = 1$.
• $y = 8 + \frac{1}{8}t$ implies $\frac{dy}{dt} = \frac{1}{8}$.

Step 3: Evaluate $T_x$ and $T_y$ at the required point

At $t = 8$:

• $x = 1 + 8 = 9$,
• $y = 8 + \frac{1}{8}(8) = 8 + 1 = 9$.

The problem states that $T_x(3, 9) = 5$ and $T_y(3, 9) = 9$. We use these values for $T_x$ and $T_y$, assuming the coordinates match at $(3, 9)$ even though the bug's path is at $(9, 9)$. The formulation suggests using these values directly.

Step 4: Calculate $\frac{dT}{dt}$

Now apply the chain rule:

$\frac{dT}{dt} = T_x(3, 9) \cdot \frac{dx}{dt} + T_y(3, 9) \cdot \frac{dy}{dt}$ $\frac{dT}{dt} = 5 \cdot 1 + 9 \cdot \frac{1}{8}$ $\frac{dT}{dt} = 5 + \frac{9}{8} = 5 + 1.125 = 6.125$

Final Answer:

The rate at which the temperature is rising after 8 seconds is approximately $\boxed{6.13} \, \text{°C/s}$.

Would you like more details or have any questions?

Here are 5 related questions you might explore:

1. How does the chain rule apply to other multivariable functions in physics?
2. Can you find the position of the bug at any arbitrary time $t$?
3. What happens to the temperature rate if $T_x$ or $T_y$ change over time?
4. How would you solve this if the temperature function were known explicitly?
5. Can you derive a general formula for $\frac{dT}{dt}$ along any arbitrary path?

Tip: When dealing with multivariable calculus, always carefully apply the chain rule, as it helps in linking rates of change across multiple variables.