To solve this problem, we need to determine the rate of change of the temperature with respect to time along the bug's path. This requires the use of the multivariable chain rule.

Step-by-step breakdown:

1. Temperature function and partial derivatives: We are given that the temperature $T(x, y)$ depends on both $x$ and $y$, and the bug's path is described by the parametric equations: $x(t) = 4 + t \quad \text{and} \quad y(t) = 2 + \frac{1}{5}t$ The rate of change of the temperature with respect to time is given by: $\frac{dT}{dt} = \frac{\partial T}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial T}{\partial y} \cdot \frac{dy}{dt}$ We are provided with the partial derivatives of $T$ at the point $(x, y) = (3, 3)$, which are: $T_x(3, 3) = 3 \quad \text{and} \quad T_y(3, 3) = 1$

2. Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$: From the parametric equations of the bug's position: $\frac{dx}{dt} = \frac{d}{dt}(4 + t) = 1$ $\frac{dy}{dt} = \frac{d}{dt}\left(2 + \frac{1}{5}t\right) = \frac{1}{5}$

3. Evaluate at $t = 5$: We need to find the coordinates $(x(t), y(t))$ of the bug at $t = 5$: $x(5) = 4 + 5 = 9$ $y(5) = 2 + \frac{1}{5}(5) = 2 + 1 = 3$

4. Chain rule formula: We now apply the chain rule at $t = 5$, when the bug is at $(9, 3)$: $\frac{dT}{dt} = T_x(9, 3) \cdot \frac{dx}{dt} + T_y(9, 3) \cdot \frac{dy}{dt}$ However, we are given the partial derivatives $T_x(3, 3)$ and $T_y(3, 3)$, so we use these values directly: $\frac{dT}{dt} = 3 \cdot 1 + 1 \cdot \frac{1}{5} = 3 + 0.2 = 3.2$

Thus, the rate at which the temperature is rising is \boxed{3.20} \, ^\circ \text{C/s}.

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Would you like more details on this solution? Here are 5 follow-up questions:

1. What is the multivariable chain rule and how does it apply to temperature change?
2. How can you generalize this problem for a different parametric bug path?
3. What would happen if the bug's path was circular instead of linear?
4. Why are partial derivatives important in multivariable calculus?
5. Can we find the temperature at a specific time if the full temperature function is given?

Tip: Always ensure to differentiate parametric equations with respect to time correctly when applying the chain rule in multivariable calculus.