Question

The slope \(S\) of a beach is related to the average diameter \(d\) (in millimeters) of the sand particles on the beach by the equation \(S=0.159+0.118 \log d\). Describe the transformation of \(f(d)=\log d\) represented by \(S\). Then use the function to determine the slope of a beach for each sand type below. $$ \begin{array}{|c|c|} \hline \text { Sand particle } & \text { Diameter (mm), } d \\ \hline \text { fine sand } & 0.125 \\ \hline \text { medium sand } & 0.25 \\ \hline \text { coarse sand } & 0.5 \\ \hline \text { very coarse sand } & 1 \\ \hline \end{array} $$

Short answer

Upon substituting the respective diameter values in the function S = 0.159 + 0.118 \log d, we can find the slope for each sand type. For fine sand it's approximately 0.21, for medium sand it's approximately 0.28, for coarse sand it's approximately 0.34 and for very coarse sand it's 0.159 since log(1) equals 0. More precise figures can be obtained using a calculator.

Step-by-step solution

Identify the transformation

The original function is f(d)=\log d. The given function S has two modifications to it, a vertical shift and a vertical stretch. The '+ 0.159' represents a vertical shift 0.159 units upwards. The '0.118' preceding the \log d represents a vertical stretch of 0.118. Meaning, it affects the 'steepness' of the log curve.

Evaluate the function for fine sand

The diameter \(d\) for fine sand is 0.125 mm. Substitute 0.125 for \(d\) in the equation S = 0.159 + 0.118 \log d: S = 0.159 + 0.118 \log(0.125). Calculate the value to find the slope for fine sand.

Evaluate the function for medium sand

The diameter \(d\) for medium sand is 0.25 mm. Substitute 0.25 for \(d\) in the equation S = 0.159 + 0.118 \log d: S = 0.159 + 0.118 \log(0.25). Calculate the value to find the slope for medium sand.

Evaluate the function for coarse sand

The diameter \(d\) for coarse sand is 0.5 mm. Substitute 0.5 for \(d\) in the equation S = 0.159 + 0.118 \log d: S = 0.159 + 0.118 \log(0.5). Calculate the value to find the slope for coarse sand.

Evaluate the function for very coarse sand

The diameter \(d\) for very coarse sand is 1 mm. Substitute 1 for \(d\) in the equation S = 0.159 + 0.118 \log d: S = 0.159 + 0.118 \log(1). Calculate the value as it is equal to the base of the logarithm to find the slope for very coarse sand.

Key concepts

Logarithmic Functions

Logarithmic functions are a crucial part of mathematics. They help us understand how things grow or shrink exponentially. In the problem provided, we start with a basic logarithmic function: \( f(d) = \log d \). This function takes the log of the diameter of sand particles. The logarithm here indicates how many times we must multiply the base, usually 10, to get the diameter. This is helpful in modeling because it captures growth with a rate that changes depending on the value of \(d\). By transforming \( \log d \) with coefficients, we can model real-world situations, such as the slope of a beach. This transformation changes the original curve, making it useful for specific scenarios or data.

Vertical Shifts

Vertical shifts in functions represent moves up or down on the graph. For the equation given, \( S = 0.159 + 0.118 \log d \), the term \(+ 0.159\) is responsible for the vertical shift. It takes your original function \( f(d) = \log d \) and moves it 0.159 units upward.

• This means every point on the logarithmic curve \( \log d \) shifts up by 0.159 units.
• This shift doesn't change the shape of the curve, merely its position along the y-axis.

As a result, the function value for any given \( d \) is increased by 0.159, reflecting how the overall slope's baseline has changed in the context of the problem.

Vertical Stretch

A vertical stretch impacts the steepness or rate of change of a function's graph. In the given function \(S = 0.159 + 0.118 \log d\), the coefficient \(0.118\) is responsible for this transformation. It modifies how stretched or compressed the curve of the function \(\log d\) becomes.

• With the coefficient being less than 1, here 0.118, every output value of \(\log d\) is multiplied by 0.118. This makes the curve less steep.
• This stretch doesn't affect the horizontal positioning of the graph but changes how the function values rise or fall.

The vertical stretch helps in determining the specific scenario where small changes in \(d\) might not significantly alter \(S\), representing the slope of the beach.

Slope Calculation

Slope calculation is about finding the rate at which elevation changes over a distance. Here, you’re asked to calculate the beach’s slope for different sand particle sizes using the transformed function \( S = 0.159 + 0.118 \log d \).

One begins by inserting specific values of \(d\), the sizes given for fine, medium, coarse, and very coarse sand particles, into the formula:

• For fine sand \(d = 0.125\), find \( S \) by calculating \(0.159 + 0.118 \log(0.125)\).
• Next, for medium sand \(d = 0.25\), calculate \(0.159 + 0.118 \log(0.25)\).
• Repeat for coarse sand \(d = 0.5\) and very coarse sand \(d = 1\).

The slope, \(S\), changes based on \(d\) because the logarithmic operation compresses growth, each result demonstrating how the size of sand particles impacts the beach slope. This helps to understand real-life applications, such as designing structures on beaches or studying erosion patterns.