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 \(\frac{1}{2}\) foot per second.

Short answer

0.0212 inches

Step-by-step solution

Understand the given stream speed

The speed of the stream is mentioned as \(\frac{1}{2}\) foot per second. This value will be substituted in the formula for the particle size that the stream can carry.

Substitute in the formula

The formula given is for the size of the largest particle a stream can carry is \(0.03 \sqrt{v}\) inches. Substitute the value of the speed \(v\) which is \(\frac{1}{2}\) into this formula to get \(0.03 \sqrt{\frac{1}{2}}\) inches.

Simplify the mathematical expression

Compute the square root of \(\frac{1}{2}\) which is approximately 0.7071, then multiply this by 0.03 to get approximately 0.0212. Hence, the size of the largest particle that the stream can carry is 0.0212 inches.

Key concepts

Stream Velocity Particle Size Relationship

The concept of stream velocity particle size relationship is rooted in the study of geomorphology and sediment transport. Essentially, it tells us how the speed of a stream affects the size of the particles that the water can move. This relationship is critical for understanding erosion and sedimentation patterns in natural water systems.

The faster a stream flows, the larger the particles it can carry. This is because increased velocity provides more energy to lift and transport sediment. In our exercise, the relationship is represented by the formula for the size of the largest particle a stream can carry, which is expressed as a function of the stream’s velocity \(v\). The formula, \(0.03 \sqrt{v}\) inches, indicates that particle size is directly proportional to the square root of the velocity. Thus, as velocity increases, the square root increases at a slower rate than the velocity, but the capacity to carry larger particles still increases.

When solving problems like these, it’s crucial to consider factors such as the shape and density of the particles, the viscosity of the water, and the slope of the stream bed, as they can all influence the stream's carrying capacity.

Solving Square Root Applications

Working with square roots is a fundamental concept in algebra that often pertains to real-world applications such as the rate of erosion in streams. Understanding how to handle square roots allows for the solution of equations where the variable is under a square root sign or when determining the dimensions of an object, such as the size of particles in our exercise.

To solve the square root application in the given problem, we start by identifying that we need to find the square root of the velocity \(v\), which is \(\frac{1}{2}\) foot per second in this case. The square root of \(\frac{1}{2}\) is about 0.7071, and we arrived at this by either using a calculator or by recognizing that the square root of \(\frac{1}{2}\) is the same as the square root of 1 divided by the square root of 2, which simplifies to \(\frac{1}{\sqrt{2}}\), or approximately 0.7071. After calculating the square root, we then proceed to the next algebraic steps, which involve multiplication or division to solve for the desired variable.

Applying Formulas in Algebra

Formulas are the bread and butter of algebra, providing a structured method to solve a wide array of problems. In our context, the formula \(0.03 \sqrt{v}\) is used to calculate the largest size of particles that a stream can carry given the velocity \(v\).

Applying formulas requires a systematic approach: first, identify the unknown variable you’re solving for, and then substitute known values for the other variables in the formula. In the erosion problem, we want to find the largest particle size, and the known variable is the stream velocity \(v\). After substituting the known value into the formula, the next step is simplifying the equation through arithmetic operations. In our case, we calculated the square root of the velocity, and then multiplied the result by 0.03 to find the particle size. This process exemplifies how algebraic formulas are applied in real-world scenarios, turning abstract mathematical operations into practical solutions.