Problem 31 - 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 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: \(f(x)=\frac{3}{400} x(40-x)\) (a) A boat proceeds at a constant speed of 5 \(\mathrm{m} / \mathrm{s}\) from a point \(A\) on the west bank while maintaining a heading perpendicular to the bank. How far down the river on the opposite bank will the boat touch shore? Graph the path of the boat. (b) Suppose we would like to pilot the boat to land at the point \(B\) on the east bank directly opposite \(A\) . If we maintain a constant speed of 5 \(\mathrm{m} / \mathrm{s}\) and a constant heading, find the angle at which the boat should head. Then graph the actual path the boat follows. Does the path seem realistic?

Step-by-Step Solution

Answer

Boat lands approximately 80 meters downstream; Angle needed depends on counteracting river speed.

Step 1: Understanding the Problem

We are given a river model where the water speed varies as a quadratic function: \( f(x) = \frac{3}{400} x(40-x) \). The boat travels from point A (west bank) at a speed of 5 m/s perpendicular to the river. We need to determine how far down the river the boat lands on the opposite bank and the impact of the river's flow on the boat's path.

Step 2: Analyzing Boat Speed Against River Flow

The boat's speed is constant at 5 m/s, perpendicular to the river, and is unaffected in the north-south direction by the river's current because the function \( f(x) \) shows east-west velocity. The boat should thus reach the opposite bank 40 m away in \( \frac{40}{5} = 8 \) seconds.

Step 3: Calculating Boat Drift Due to Current

To find the downstream position due to river current, we need the integral of water speed over time. Integrate the current speed function \( f(x) = \frac{3}{400} x(40-x) \) from west to east over 8 seconds.

Step 4: Integral of Water Speed Over Time

Assume the boat drifts uniformly over the 40-meter breadth. Position \(x\) thus changes linearly from 0 to 40 at constant speed. In 8 seconds, calculate \( \int_{0}^{8} f(x(t)) dt \), assuming linear drift rate: \( x(t) = \frac{40}{8}t = 5t \). So, \[ x(t) = 5t \] meaning \( f(x(t)) = \frac{3}{400}(5t)(40-5t) \).

Step 5: Evaluate the Integral

Evaluate the expression: \( \int_{0}^{8} \frac{15}{400}t(40-5t) dt \). Simplify and solve the integral using substitution: \( \frac{15}{400} \int_{0}^{8} (200t - 25t^2) dt \). Calculate \[ \frac{15}{400} \left[ 100t^2 - \frac{25}{3}t^3 \right]_{0}^{8} \].

Step 6: Finish Integral Evaluation

Plug in limits to get: \( \frac{15}{400} \left[ (100 \times 64) - \frac{25 \times 512}{3} \right] \). Simplify further: \( \frac{15}{400} (6400 - 4266.67) = \frac{15}{400} (2133.33) \approx 80 \). The boat lands 80 m downstream.

Step 7: Calculating Heading Angle for Point B

To reach directly across at point B, calculate the angle \( \theta \) relative to the perpendicular needed to counteract the current. Set east-west speed from river equal to lateral boat progress: \( v_r = 5 \sin\theta \) (because north is \(5 \cos\theta\)).

Key Concepts

Quadratic Function

Quadratic functions are an essential part of algebra that help in modeling various real-world situations. In this exercise, the river's current is represented as a quadratic function, specifically given by \( f(x) = \frac{3}{400} x(40-x) \). This function describes how the speed of the river changes depending on the distance \( x \) from the west bank.
The quadratic function in this context is used because it provides a parabolic curve, which naturally fits the scenario where the flow is fastest in the middle and slowest at the edges. It is structured in the form \( ax^2 + bx + c \), where the highest power of \( x \) dictates that it is a quadratic. Evaluating the function's values at selected points can help students visualize how the speed varies.

• At \( x = 0 \) (west bank), the speed is 0.
• At \( x = 40 \) m (east bank), the speed is also 0.
• In the middle, at \( x = 20 \) m, the speed reaches its maximum.

This characterization is helpful when analyzing river flows or similar phenomena where rates vary predictably along a specific segment. By comprehending how quadratic functions can represent these changing rates, students gain a valuable tool in modeling natural and engineered systems.

Integration

Integration is a fundamental concept in calculus used to calculate the area under a curve, which, in one-dimensional motion problems, can represent the total distance moved given a varying velocity. In this problem, we calculated how far downstream a boat would drift due to the river current during its crossing. The water speed as a function of river width is given by a quadratic function, and integrating this function from 0 to the time taken to cross (8 seconds here) allows us to find the total downstream distance.
The following steps were taken for the integration:

• We considered the quadratic function of the river's velocity \( f(x) = \frac{3}{400} x(40-x) \).
• Substituting the changing position \( x(t) = 5t \), where \( 5t \) represents the linear eastward progression of the boat.
• Set up the integral: \( \int_{0}^{8} \frac{15}{400}t(40-5t) dt \).
• Evaluated the definite integral to find the total drift. This required first simplifying the expression and then finding the antiderivative.
  

Integration helps us find not just how far, but also how these distances and functions relate over time, enhancing understanding of motion in varying systems such as turbulent river currents.

River Current Model

The river current model used here is apt for simulating real river flows where water speed varies across the width of the river. Named as a quadratic model, it elucidates how such variations can be mathematically articulated. In the real world, water flow is affected by numerous factors: channel shape, bed roughness, and obstructions.
In the exercise, the river's width is 40 meters, and maximum velocity is given at the center using the quadratic function \( f(x) = \frac{3}{400} x(40-x) \). The river is fastest at the midpoint due to the quadratic nature of the formula.
The usage of a quadratic model is particularly beneficial because:

• It accounts for velocity differences between the center and banks effectively.
• Provides a simple yet effective representation of potentially complex natural processes.

Thus, using a quadratic equation emphasizes how mathematical functions can be part of hydrodynamic modeling, helping predict water behavior, vital for boat navigation and related calculations.

Boat Trajectory

Boat trajectory under the influence of a river current is an interesting application of physics and mathematics. The trajectory tells us about the path the boat will follow due to its own velocity and additional factors like the river current. In this problem, the boat starts from point A on the west bank and aims to land directly opposite on the east bank, maintaining a speed of 5 m/s.
Two important considerations are:

• When the boat aims perpendicular to the river, it is impacted by the current and lands downstream.
• When aiming to land at a specific point opposite, the heading and angle must counteract the river's drift in the east-west direction.

To achieve a direct crossing, we calculated the angle \( \theta \) which counteracted current drift. Using trigonometric relationships:

• The north-south velocity requires setting the east-west drift component \( v_r \) to match the lateral movement at \( 5 \sin\theta \).

Graphing these trajectories can help visualize the effect of current on the motion, making it excellent for understanding forces in real-world vector navigation.