A man is on the bank of a river that is 1 mile wide. He...

A man is on the bank of a river that is 1 mile wide. He wants to travel to a town on the opposite bank, but 1 mile upstream. He intends to row on a straight line to some point $P$ on the opposite bank and then walk the remaining distance along the bank (Figure $\mathrm{Ex}-56$ ). To what point should he row in order to reach his destination in the least time if (a) he can walk $5 \mathrm{mi} / \mathrm{h}$ and row $3 \mathrm{mi} / \mathrm{h}$ (b) he can walk $5 \mathrm{mi} / \mathrm{h}$ and row $4 \mathrm{mi} / \mathrm{h} ?$

A man is on the bank of a river that is 1 mile wide. He wants to travel to a town on the opposite bank, but 1 mile upstream. He intends to row on a straight line to some point $P$ on the opposite bank and then walk the remaining distance along the bank (Figure $\mathrm{Ex}-56$ ). To what point should he row in order to reach his destination in the least time if
(a) he can walk $5 \mathrm{mi} / \mathrm{h}$ and row $3 \mathrm{mi} / \mathrm{h}$
(b) he can walk $5 \mathrm{mi} / \mathrm{h}$ and row $4 \mathrm{mi} / \mathrm{h} ?$

Key Concepts

Differentiation and Critical Point Analysis

Differentiation is used to compute the rate of change of the function that models the total time. By setting the derivative to zero, one can solve for the critical points at which the function may reach a minimum (or maximum). This process, including checking that the solution indeed minimizes the objective function, is at the core of solving many optimization problems in calculus.

Euclidean Geometry and the Distance Formula

The distance formula, based on the Pythagorean theorem, calculates the straight-line distance between two points in a plane. In optimization problems involving travel across regions—for example, crossing a river or moving along different paths—this geometric relationship is used to express one segment of the journey (typically the diagonal or the direct path) in terms of other variables, forming part of the total distance or time function.

Calculus-Based Optimization

Optimization in calculus involves finding the extreme values of a function, such as a minimum or maximum, by setting up a mathematical model and then using techniques like differentiation. In problems where multiple modes of travel or actions are combined, an objective function (for example, total time) is expressed in terms of a variable that can be adjusted, and then calculus is used to identify the point at which that function is minimized.

Rate, Distance, and Time Relationship

The relationship between rate, distance, and time is fundamental in problems related to motion, where time is determined by dividing the distance traveled by the speed. When different segments of a journey are undertaken at different speeds, it becomes necessary to model each segment's travel time separately and then add these to form the total time, which is then optimized.

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