Problem 29 - Essential Calculus Early Transcendentals

Question

Water traveling along a straight portion of a river normally flows fastest in the middle, and the speed slows to almost zero at the banks. Consider a long straight stretch of river flowing north, with parallel banks 40 \(\mathrm{m}\) apart. If the maximum water speed is \(3 \mathrm{m} / \mathrm{s},\) we can use a quadratic function as a basic model for the rate of water flow \(x\) units from the west bank: \(f(x)=\frac{3}{400} x(40-x)\) $$\begin{array}{l}{\text { (a) A boat proceeds at a constant speed of } 5 \mathrm{m} / \mathrm{s} \text { from a }} \\ {\text { point } A \text { on the west bank while maintaining a heading }} \\ {\text { perpendicular to the bank. How far down the river on }}\end{array}$$$$ \begin{array}{l}{\text { the opposite bank will the boat touch shore? Graph the }} \\ {\text { path of the boat. }}\end{array}$$ $$\begin{array}{l}{\text { (b) Suppose we would like to pilot the boat to land at the }} \\ {\text { point } B \text { on the east bank directly opposite } A . \text { If we }} \\ {\text { maintain a constant speed of } 5 \mathrm{m} / \mathrm{s} \text { and a constant }} \\ {\text { heading, find the angle at which the boat should head. }} \\ {\text { Then graph the actual path the boat follows. Does the }} \\ {\text { path seem realistic? }}\end{array}$$

Step-by-Step Solution

Answer

(a) The boat lands 12 m downriver. (b) Head upstream at 17.46° to land directly across.

Step 1: Understanding the Boat's Path

The boat moves from west to east, perpendicular to the river banks. The river flows north, which would cause the boat to move at an angle downriver. The speed of the river varies, being maximum in the middle and zero at the banks. This speed profile affects the path the boat takes.

Step 2: Calculate Effective Downstream Speed

The boat's speed perpendicular to the bank is 5 m/s. The river's flow affects the boat's trajectory. On average, the river speed across the entire width can be calculated as the average of the quadratic speed profile: \\[f(x) = \frac{3}{400} x(40-x)\].\ At x=20, the speed is maximal (1.5 m/s). Thus, the effective downstream speed is approximated by averaging the river flow speed over the width.

Step 3: Determine Distance Downriver

Calculate the drift caused by the river flow. The boat's speed across the river is 5 m/s over a distance of 40 m. Time to cross = Distance/Speed = 40 m / 5 m/s = 8 s. During this time, calculate how far the boat is carried downstream by the average flow of the river. If the average river speed is estimated at 1.5 m/s, then downstream drift = 1.5 m/s * 8 s = 12 m.

Step 4: Graph the Path of the Boat

Plot the path considering the initial perpendicular direction and the downstream drift. Start from point A (0,0) and plot an endpoint at (40,12) to reflect 12 meters drift by the river. The path would appear as a straight line slanted downstream from west to east bank.

Step 5: Find Angle for Perpendicular Landing

To reach point B directly across from A, the boat must head upstream at an angle, counteracting the river's drift. If the speed of the boat is 5 m/s, and we must counteract an average downstream speed of 1.5 m/s, adjust the heading angle using simple vector addition to solve for the correct angle.

Step 6: Calculate Heading Angle

The heading angle θ can be found using the equation \( \sin(\theta) = \frac{1.5}{5}\). Solve for θ: \( \theta = \arcsin\left(\frac{1.5}{5}\right) \approx 17.46^\circ \). This is the angle at which the boat must head upstream to reach the opposite bank directly across.

Step 7: Graph the Corrected Path

Graph the path by plotting the vector with calculated heading, leading to a path from (0,0) directly perpendicular to the opposite bank, correcting for river drift. The path should start from A (0,0) and end at B (40,0).

Key Concepts

Boat Trajectory Calculations

Calculating a boat's trajectory across a flowing river involves understanding both the boat's movement and the influence of the river's flow. Imagine steering a boat from one side of a 40-meter-wide river to another, aiming directly across. The boat travels at a speed of 5 meters per second towards the opposite bank. However, the river itself has a current that affects the boat's path.To begin, consider the river’s speed profile, which is modeled by a quadratic function. This function describes how the river's speed changes from slow near the banks to fastest in the middle. This forces the boat to follow a trajectory not quite perpendicular with the bank, resulting in a drift downstream.Key factors in this calculation include:

• The boat's speed perpendicular to the bank (5 m/s).
• The river's speed as a function: \(f(x) = \frac{3}{400} x(40 - x)\).

Together, these factors help determine how far down the riverbank the boat will touch shore. This introduces the concept of drift caused by the river's current.

Perpendicular Speed and Drift

When considering the impact of the river's current on the boat's course, it's essential to examine the concept of speed and drift. The perpendicular speed of 5 m/s that the boat maintains refers to its motion directly across the river, not considering the river's flow. However, the river adds a horizontal drift component to the boat's movement. The speed isn’t uniform across; it varies according to the quadratic function previously described: fastest at the middle, slower at the banks. This average speed affects how much the boat will drift downstream while crossing. To calculate how much the river’s current affects the boat:

• Determine the average speed along the river width, thanks to the quadratic profile.
• The time taken for the boat to cross (40m / 5m/s = 8s).
• Estimate the downstream drift: average river speed (1.5 m/s) multiplied by crossing time: 12 meters drift.

This calculation shows how far the boat is swept downstream due to the river current before reaching the opposite bank.

Graphing River and Boat Paths

To visualize the impact of the river’s drift on a boat's path, graphing provides valuable insights. Initially, you’d draw the intended path: a straight line from point A on the west bank to directly across the east bank. However, due to the river's current, this path curves downstream. To represent this in a graph:

• Start at the point of origin, A (0,0).
• Consider the effect of river flow, leading to an endpoint at (40,12) to account for drift.
• The path results in a slanted line from west to east, illustrating downstream displacement.

Now, if the aim is to reach directly across, corrections in the heading must be made. By finding and graphing a path at an adjusted angle — previously calculated using sine trigonometric functions — the boat's intended destination can be achieved, counteracting the drift. This demonstrates the necessity of adjusting the trajectory to land directly opposite the starting point regardless of the river's movement.