a boat on the ocean is 2 mi from the nearest point on a straight shoreline; that point is 13 mi from a…

Explanation:

Step1: Let the distance from the landing - point to the restaurant be $x$ miles.

The distance rowed $d_1=\sqrt{2^{2}+(13 - x)^{2}}=\sqrt{4+(13 - x)^{2}}$ and the distance walked is $x$. Let the rowing speed be $r$ and the walking speed $w = 3$ mi/hr. The total time $T=\frac{\sqrt{4+(13 - x)^{2}}}{r}+\frac{x}{3}$. When rowing directly to the restaurant ($x = 13$), we want this to be the minimum - time path.

Step2: Differentiate the time function with respect to $x$.

The derivative of $T$ with respect to $x$ is $T^\prime=\frac{-(13 - x)}{r\sqrt{4+(13 - x)^{2}}}+\frac{1}{3}$. When rowing directly to the restaurant ($x = 13$), for this to be the minimum - time path, the derivative of the time function with respect to $x$ at $x = 13$ must be non - negative. In the general case of minimizing time, we set $T^\prime = 0$ for other values of $x$. When rowing directly ($x = 13$), we consider the ratio of distances and speeds. The straight - line distance from the boat to the restaurant is $d=\sqrt{2^{2}+13^{2}}=\sqrt{4 + 169}=\sqrt{173}$ miles.

Step3: Use the fact that for the direct path to be the quickest.

The time taken to row directly $t_1=\frac{\sqrt{173}}{r}$ and the time taken if we consider a non - direct path (using the general time formula). If the direct path is the quickest, we can use the fact that when we compare the time of rowing directly and the time of rowing to a point and then walking. Let's consider the relationship between the distances and speeds. The distance of walking is $x$ and rowing is $\sqrt{4+(13 - x)^{2}}$. When rowing directly, we know that the ratio of distances and speeds should satisfy the condition for minimum time. The distance of rowing directly is $\sqrt{4 + 13^{2}}=\sqrt{173}$ miles and the distance of walking if not rowing directly is non - zero. If we consider the time formula $T=\frac{d_{row}}{r}+\frac{d_{walk}}{w}$, for the direct path ($d_{walk}=0$) to be the quickest, we use the fact that the speed of walking $w = 3$ mi/hr. We know that if we consider the right - triangle with sides 2 and 13, the hypotenuse $D=\sqrt{2^{2}+13^{2}}=\sqrt{173}$. The time taken to walk the maximum possible non - zero distance (if not rowing directly) and rowing a non - direct path should be greater than the time taken to row directly. Let the time of rowing directly $T_{direct}=\frac{\sqrt{173}}{r}$ and the time of rowing to a point and then walking. If we consider the limit case, we know that the ratio of the distance of rowing directly to the distance of walking (if we were to use a non - direct path) and their respective speeds. The distance of rowing directly is $\sqrt{173}$ miles and the maximum distance of walking is 13 miles. We want $\frac{\sqrt{173}}{r}\leq\frac{\sqrt{173}}{3}$ (when considering the time comparison for the direct path to be the quickest). The minimum speed $r$ for which rowing directly is the quickest is given by considering the ratio of the hypotenuse of the right - triangle (formed by the boat's position, the nearest point on the shore, and the restaurant) to the side along the shore. The distance from the boat to the restaurant is $\sqrt{2^{2}+13^{2}}=\sqrt{4 + 169}=\sqrt{173}$ miles. We know that for the direct path to be the quickest, the ratio of speeds should satisfy the condition based on the distances in the right - triangle. The minimum speed $r$ of rowing such that rowing directly is the quickest is $r=\frac{3\sqrt{173}}{13}$ mi/hr.

Answer:

$\frac{3\sqrt{173}}{13}$