Problem 40 - Calculus Single Variable

Question

A boat starts for the opposite shore of a \(2 \mathrm{~km}\) wide river so that (a) the boat always heads toward the point that is directly across from the original launch point, and (b) the speed of the boat is three times faster than the current of the river. If we set up on an \(x y\) -coordinate system so that the boat's initial position is \((2,0),\) the landing site is at (0,0) , and the current is in the positive \(y\) -direction, then the path of the boat is given by $$ y=\left(\frac{x}{2}\right)^{2 / 3}-\left(\frac{x}{2}\right)^{4 / 3} $$. What are the \(x\) coordinates at the two points at which the boat is \(0.137 \mathrm{~km}\) downstream.

Step-by-Step Solution

Answer

The \(x\) coordinates are approximately 0.4194 km and 1.5806 km.

Step 1: Understand the Equation of the Boat's Path

The path of the boat is given as \( y = \left(\frac{x}{2}\right)^{2/3} - \left(\frac{x}{2}\right)^{4/3} \). This equation describes how the boat moves across the river considering both its own speed relative to the current.

Step 2: Setting Up the Problem

We know that we need to find the values of \(x\) for which \(y = 0.137\). This will help us determine the positions downstream where the boat is 0.137 km from its starting path.

Step 3: Substitute and Rearrange Equation

Substitute \(y = 0.137\) into the path equation: \[ 0.137 = \left(\frac{x}{2}\right)^{2/3} - \left(\frac{x}{2}\right)^{4/3} \]. Simplifying it: \[ 0.137 = u^{2/3} - u^{4/3} \] where \(u = \frac{x}{2}\).

Step 4: Solve for \(x\) Using a Substitution

Let \(u = \left(\frac{x}{2}\right)\), therefore: \[ u^{2/3} - u^{4/3} = 0.137 \]. From this equation, we find the values of \(u\) that satisfy it, which will consequently help us find \(x\).

Step 5: Numerical Solution for \(u\)

Using numerical methods or graphing calculator, solve for \(u\) in the equation \( u^{2/3} - u^{4/3} = 0.137 \). Approximate solutions are \(u_1 \approx 0.2097\) and \(u_2 \approx 0.7903\).

Step 6: Determine \(x\) from \(u\)

Recalling that \(u = \frac{x}{2}\), solve for \(x\) for each \(u\): If \(u_1 = 0.2097\), then \(x_1 = 2 \times 0.2097 = 0.4194\). If \(u_2 = 0.7903\), then \(x_2 = 2 \times 0.7903 = 1.5806\).

Step 7: Final Answer

The boat is 0.137 km downstream with the \(x\)-coordinates approximately \(x_1 \approx 0.4194\text{ km}\) and \(x_2 \approx 1.5806\text{ km}\).

Key Concepts

Boat Navigation Path

In this exercise, we explore the path of a boat crossing a river, which is influenced by both its speed and the river's current. When navigating such a path, it's essential to consider that the boat aims for a target directly opposite its starting point. This is a common scenario in river crossing problems where the angle of the boat's travel path compensates for the flow of the river.

• The boat's velocity forms a resultant vector that combines its speed and the current's direction. Here, the additional complexity arises because the boat travels three times faster than the current.
• As it heads towards its target, the set of possible paths can sometimes be represented by particular equations like the one given in our problem: \[ y = \left(\frac{x}{2}\right)^{2/3} - \left(\frac{x}{2}\right)^{4/3} \]

This equation helps visualize how the boat advances along the river despite the force opposing its progress.

River Crossing

Crossing a river involves careful navigation due to the presence of water currents that can shift an object's course. This exercise focuses on a boat that moves against a current leading to two crucial navigation elements:

• Current Influence: The water flow affects how a boat moves across the river, making it drift from its intended straight path unless compensated by an increased speed or adjusted angle.
• Target Orientation: To reach a specific point directly across, the boat must adjust its navigation to account for the current's strength and direction.

Here, the boat aims directly across a \(2\,\text{km}\) wide river, with its speed counterbalancing the downstream force, allowing it to maintain a path largely directed toward its final position.

Numerical Methods

When solving equations with variables that are not easily simplified, numerical methods are a valuable approach. These methods use computational algorithms to find approximate solutions, especially for problems involving non-linear equations like in our exercise.

• Problem Setup: We set up the equation \[ 0.137 = \left(\frac{x}{2}\right)^{2/3} - \left(\frac{x}{2}\right)^{4/3} \] to find values of \(x\) where the boat is downstream by \(0.137\, \text{km}\).
• Substitution Technique: Introducing a new variable \(u = \frac{x}{2}\) simplifies the computation, transitioning the equation into \[ u^{2/3} - u^{4/3} = 0.137 \].
• Solver or Graphing: Utilizing software for numerical approximation or visual graph analysis is essential here, providing solutions \(u_1 \approx 0.2097\) and \(u_2 \approx 0.7903\).

By transforming and solving the equation numerically, we can estimate the exact points where the path condition is satisfied.

Equation Solving

Equations like the one in this exercise require solving techniques that accommodate non-linear terms. Here, we need to zero in on the specific values of \(x\) where our path equation holds true.

• Essential Steps: We substitute known values into the equation, \[ y = \left(\frac{x}{2}\right)^{2/3} - \left(\frac{x}{2}\right)^{4/3} \], setting \(y = 0.137\) to find the intersections or solution points.
• Rearranging Terms: Simplifying gives \[ 0.137 = u^{2/3} - u^{4/3} \], allowing us to solve for the new variable \(u\).
• Back-substitution: Once \(u\) values are determined, we conclude by calculating \(x\) via the relationship \(u = \frac{x}{2}\), resulting in \(x_1 \approx 0.4194\, \text{km}\) and \(x_2 \approx 1.5806\, \text{km}\).

Each step in solving ensures clarity and accuracy in determining the boat's position downstream.