The solution to the problem involves determining the optimal location for landing the boat and the minimum rowing speed required for the shortest time.

Part (a): Finding the Optimal Landing Point

1. Set Up Variables and Function for Total Time:

   • Let $x$ be the distance from point $A$ (the nearest shore point to the boat) to the landing point on the shore.
   • The distance she rows to the shore is $\sqrt{x^2 + 9}$, and the distance she walks to the restaurant is $9 - x$.

2. Time Function:

   • The total travel time $T(x)$ is given by the sum of the time rowing and walking: $T(x) = \frac{\sqrt{x^2 + 9}}{2} + \frac{9 - x}{3}$
   • This equation combines both the rowing and walking segments based on their respective speeds.

3. Find Critical Points:

   • Differentiate $T(x)$ with respect to $x$, set $T'(x) = 0$, and solve for $x$.
   • In this case, the solution yields $x = \frac{6\sqrt{5}}{5}$ miles from point $A$, or equivalently, $9 - \frac{6\sqrt{5}}{5}$ miles from the restaurant.

Part (b): Minimum Rowing Speed for Direct Route

1. Set Up Condition for Direct Rowing:

   • For the fastest route directly to the restaurant, we want to set $x = 9$, and we solve for the minimum rowing speed, $r$, so that rowing directly takes less time than the combined rowing and walking.

2. Determine Minimum Rowing Speed:

   • Solving with given conditions, we find that $r = \frac{15}{\sqrt{29}} \approx 2.79$ mi/hr.

This approach should match the given answers in the problem. Let me know if you need more clarification or a step-by-step breakdown.

Would you like more details on a specific step or concept? Here are related questions to deepen understanding:

1. How does changing the walking speed affect the optimal landing point?
2. What is the significance of the critical point in optimization problems like this?
3. How would the time equation change if the rowing speed were faster than the walking speed?
4. Can this method be applied to other similar distance minimization problems?
5. What would happen if the destination were at a different point on the shore?

Tip: In optimization problems, setting up equations carefully and identifying constraints help streamline the solution process.