Problem 98 - College Algebra and Calculus: An Applied Approach

Question

Erosion A stream of water moving at the rate of \(v\) feet per second can carry particles of size \(0.03 \sqrt{v}\) inches. Find the size of the largest particle that can be carried by a stream flowing at the rate of \(\frac{7}{9}\) foot per second.

Step-by-Step Solution

Answer

The stream moving at \(\frac{7}{9}\) foot per second can carry particles of size up to \(0.01 \sqrt{7}\) inches.

Step 1: Identify the given speed of the water stream

In this problem, we are given that the speed of the stream i.e., \(v\) is \(\frac{7}{9}\) foot per second.

Step 2: Substitute the given value of 'v' in the equation

We now substitute the value of \(v\) in the given size equation. The equation for the size of particles the stream can carry is \(0.03 \sqrt{v}\). So replacing \(v\) with \(\frac{7}{9}\) gives us \(0.03 \sqrt{\frac{7}{9}}\).

Step 3: Calculate the size of the largest particle

The size of the largest particle the stream can carry is found by simplifying \(0.03 \sqrt{\frac{7}{9}}\). When we simplify this equation we get \(\frac{0.03}{3} \sqrt{7} = 0.01 \sqrt{7}\).

Key Concepts

Particle Size Calculation

One key aspect of understanding how erosion works, especially in streams and rivers, is knowing how to calculate the size of the particles that the water can transport. In this exercise, we learned that the size of the particle, denoted here as "particle size", can be calculated using the formula:

• Particle Size = \( 0.03 \sqrt{v} \)

where \(v\) represents the velocity of the water flow in feet per second.
The square root function shows the relationship between water speed and particle size.
This means that as the speed of the stream increases, the size of the particles that can be carried also increases.
For example, in the given problem, when the stream flows at \( \frac{7}{9} \) feet per second, the largest particle size that can be carried is computed by substituting \(v\) into the formula, resulting in \( 0.01 \sqrt{7} \) inches.
Understanding this calculation is crucial because it helps predict erosion patterns and manage sediment in waterways.

Rate of Water Flow

The rate at which water moves in rivers and streams is a fundamental concept in erosion studies. This flow rate, or velocity, is denoted by \(v\) in erosion calculations.
The significance of \(v\) is that it impacts the size of particles that the stream can carry, with faster flows capable of transporting larger particles.
This relationship can be expressed mathematically using the formula:

• Velocity = distance traveled / time taken

Calculating water flow rate accurately involves multiple factors like river gradient, channel shape, and water volume.
It's vital for environmental management to understand these dynamics to anticipate how natural and human activities might influence erosion and sediment transport.
This helps in creating solutions to mitigate erosion where necessary, ensuring balanced ecosystems and protecting human infrastructure.

Mathematical Modeling in Erosion

Mathematical modeling plays a critical role in predicting and understanding erosion dynamics. This involves using equations and formulas to simulate the processes of erosion.
These models are essential tools for scientists and engineers. They help in mapping out potential erosion scenarios by accounting for variables like water velocity, particle size, and riverbed characteristics.
In the exercise, we used a simple model to determine the particle size that a stream can carry. However, real-world applications often involve complex models that can include multiple variables and interactions.

• Benefits of these models include predicting erosion rates, planning for flood management, and designing effective environmental interventions.

By using mathematical modeling, experts can better forecast how changes in water flow, due to natural or human-made effects, might alter landscapes over time.