Problem 78 - Algebra and Trigonometry with Analytic Geometry

Question

A man in a rowboat that is 2 miles from the nearest point \(A\) on a straight shoreline wishes to reach a house located at a point \(B\) that is 6 miles farther down the shoreline (see the figure). He plans to row to a point \(P\) that is between \(A\) and \(B\) and is \(x\) miles from the house, and then he will walk the remainder of the distance. Suppose he can row at a rate of \(3 \mathrm{mi} / \mathrm{hr}\) and can walk at a rate of \(5 \mathrm{mi} / \mathrm{hr}\). If \(T\) is the total time required to reach the house, express \(T\) as a function of \(x\).

Step-by-Step Solution

Answer

The total time function is \(T(x) = \frac{\sqrt{4 + (6-x)^2}}{3} + \frac{x}{5}\).

Step 1: Identify Components

Identify the different segments of the man's journey. He rows to point \(P\) and then walks to the house, \(B\). The segment from the rowboat's starting point to \(P\) is \(2\) miles perpendicular to the shore, and \(x\) miles along the shoreline from \(B\).

Step 2: Use Pythagorean Theorem

Since the man rows to point \(P\), we need to calculate the distance he rows, which forms the hypotenuse of a right triangle. The rowboat starts \(2\) miles from the shore, and he travels \(6-x\) miles parallel to the shore:\[\text{Distance to } P = \sqrt{2^2 + (6-x)^2}\]

Step 3: Calculate Rowing Time

The rowing speed is \(3 \mathrm{mi/hr}\). The time spent rowing is the distance to \(P\) divided by rowing speed:\[T_{\text{row}} = \frac{\sqrt{2^2 + (6-x)^2}}{3}\]

Step 4: Calculate Walking Distance and Time

After reaching point \(P\), the man walks the remaining distance \(x\). The walking speed is \(5 \mathrm{mi/hr}\):\[T_{\text{walk}} = \frac{x}{5}\]

Step 5: Total Time Function

Add the rowing time and walking time to express the total time \(T\) as a function of \(x\):\[T(x) = \frac{\sqrt{2^2 + (6-x)^2}}{3} + \frac{x}{5}\]

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it can be expressed as:\[a^2 + b^2 = c^2\]where \(a\) and \(b\) are the lengths of the triangle's legs, and \(c\) is the length of the hypotenuse.
This theorem is immensely useful in various real-world applications, particularly in determining distances. In the exercise involving the rowboat, the man forms a right triangle with the vertical distance being 2 miles (distance from the shore) and the horizontal segment \((6-x)\) miles along the shore. Applying the Pythagorean theorem helps calculate the hypotenuse, that is, the actual rowing distance.
This concept is crucial as it transforms a 2D problem into something manageable, allowing us to relate different physical components such as distance and speed.

Distance-Time Relationship

The relationship between distance, speed, and time is a core concept in physics and mathematics, commonly expressed with the formula:
\[ ext{Distance} = ext{Speed} imes ext{Time}\]Rearranging the formula helps us find time:\[ ext{Time} = \frac{\text{Distance}}{\text{Speed}}\]This relationship is essential for solving problems where time is a function of distance traveled at varying speeds.
In the exercise scenario, the rowboat moves at 3 miles per hour, and after rowing, the walking speed is 5 miles per hour. Each segment of the journey can have its own time calculation:

• Rowing time: Distance to point \(P\) divided by the rowing speed.
• Walking time: Distance from \(P\) to the house \(B\) divided by walking speed.

Altogether, this relationship helps piece together the total time required for the man to reach his destination by combining the time calculations from different sections of his journey.

Right Triangles in Geometry

Right triangles are a staple of geometric studies because of their unique properties and how they simplify complex problems. A right triangle has one 90-degree angle, which brings into play the Pythagorean Theorem to find unknown side lengths and understand angles.
Understanding how right triangles work allows us to solve for distances indirectly. In the rowboat exercise, the path of the row forms one leg of a right triangle, and the distance along the shore forms the other. The solution identifies the hypotenuse, which represents the diagonal distance the man rows by applying the Pythagorean theorem calculation.
Learning right triangle geometry builds a foundation for solving optimization problems where indirect paths and constraints come into play, such as the shortest path or minimum time scenarios in physics, engineering, and everyday life.