Problem 37 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Loni is standing on the bank of a river that is 1 mile wide and wants to get to a town on the opposite bank, 1 mile upstream. She plans to row on a straight line to some point \(P\) on the opposite bank and then walk the remaining distance along the bank. To what point \(P\) should Loni row to reach the town in the shortest possible time if she can row at 4 miles per hour and walk at 5 miles per hour?

Step-by-Step Solution

Answer

Loni should row to a point 0.8 miles upstream.

Step 1: Understand the Problem

Loni needs to cross a river that is 1 mile wide and then walk 1 mile upstream to a town. She needs to minimize the time it takes by choosing the optimal point P to row to.

Step 2: Set Up Variables

Let Loni row to a point P that is x miles upstream from her starting point. Then, the direct rowing distance to point P can be found using the Pythagorean theorem as \(\text{distance} = \sqrt{1^2 + x^2}\)

Step 3: Calculate Rowing Time

The time it takes to row to point P, given Loni's rowing speed of 4 miles per hour, is: \(\text{time}_{\text{row}} = \frac{\sqrt{1 + x^2}}{4}\)

Step 4: Calculate Walking Time

Once Loni reaches point P, she will have to walk the remaining (1-x) miles upstream. Given her walking speed of 5 miles per hour, the time it takes to walk the remaining distance is: \(\text{time}_{\text{walk}} = \frac{1 - x}{5}\)

Step 5: Set Up the Total Time Function

To find the total time, add rowing time and walking time: \(\text{total time} = \frac{\sqrt{1 + x^2}}{4} + \frac{1 - x}{5}\)

Step 6: Minimize the Total Time

Take the derivative of the total time function with respect to x and set it to 0 to find the minimum time: \[ \frac{d}{dx} \(\frac{\sqrt{1 + x^2}}{4} + \frac{1 - x}{5}\) = 0\] Solve this derivative equation to find the x value for which the total time is minimum.

Step 7: Solve the Derivative Equation

After solving the derivative, the optimal x value is \(\frac{4}{5}\).

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It relates the sides of a right triangle, stating that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In formula terms: \[ a^2 + b^2 = c^2 \] In this exercise, the theorem helps to calculate the direct rowing distance from Loni’s starting point to point P on the opposite bank, forming a right triangle with the river width (1 mile) and the distance she rows upstream (x miles). The rowing distance to P is: \[ \text{distance} = \text{hypotenuse} = \sqrt{1^2 + x^2} \] This calculation is essential for understanding how far Loni must row to reach point P and minimizes the total travel time.

Time Minimization

Time minimization involves finding the quickest way to complete a task. In Loni’s case, her route includes rowing and walking, both at different speeds.

• Rowing speed: 4 miles per hour
• Walking speed: 5 miles per hour

We need to determine the optimal point P on the opposite bank that minimizes the total travel time. This involves setting up two time calculations:

• Rowing time: Determined by the rowing distance and speed
• Walking time: Determined by the remaining walking distance and speed

Combining these will give us the total time function to be minimized.

Derivative Calculation

Calculus comes into play when minimizing the total travel time. Here's a step-by-step approach:1. Define the total time function by adding the rowing time and walking time.2. Rowing time: \[ \text{time}_{\text{row}} = \frac{\sqrt{1 + x^2}}{4}\]3. Walking time:\[ \text{time}_{\text{walk}} = \frac{1 - x}{5} \]4. Total time function:\[ \text{total time} = \frac{\sqrt{1 + x^2}}{4} + \frac{1 - x}{5} \]Taking the derivative of the total time function with respect to x is necessary to find the minimum time. The derivative equation set to zero will help us solve for the optimal x value.

Function Optimization

Function optimization is the process of making a function as effective as possible. For Loni's problem, the goal is to minimize the total travel time. By solving the derivative equation:\[ \frac{d}{dx} \( \frac{\sqrt{1 + x^2}}{4} + \frac{1 - x}{5} \) = 0 \]We find the critical points of the function. Solving this specific problem, we find: \[ x = \frac{4}{5} \]This tells us that Loni should row to a point P that is 0.8 miles upstream from her starting point to minimize her travel time. This x value ensures that the total time spent on rowing and walking is the shortest possible under the given conditions.