Question

The height, \(h\) in \(\mathrm{cm}\), of an eroding sand dune as a function of year, \(t,\) is given by \(h=f(t) .\) Describe the difference between this sand dune and a second one one whose height is given by (a) \(h=f(t+30)\) (b) \(h=f(t)+50\).

Short answer

Answer: In case (a), the second sand dune behaves as if it is 30 years "younger" than the first sand dune. In case (b), the second sand dune is always 50 cm higher than the first sand dune, regardless of the specific value of time (t).

Step-by-step solution

Analyzing the effect of \(h=f(t+30)\)

First, let's look at the effect of shifting the time variable \(t\) by 30 years in the formula. This shifts the entire function to the left along the time axis by 30 years. Essentially, the height of the second sand dune at a given time would be the same as the height of the first sand dune 30 years later. For example, if we want to find the height of the second sand dune at a specific time, say \(t = 10\), then we would plug \(t=10\) into the formula \(h=f(t+30)\), which would give us the height of the first sand dune at \(t=40\), that is, 30 years later.

Summarizing the difference for case (a)

In the first case (a), \(h=f(t+30)\), the second sand dune behaves as if it were 30 years "younger" than the first sand dune. Height values of the second sand dune correspond to height values of the first sand dune from 30 years later.

Analyzing the effect of \(h=f(t)+50\)

In the second case (b), with \(h=f(t)+50\), the \(+50\) term in the formula represents a constant upward shift of the height by 50 cm. This means that the height of the second sand dune is always 50 cm higher than the first sand dune at any given time \(t\). For example, if the height of the first sand dune at \(t=10\) years is given by \(h=f(10)\), then the height of the second sand dune at the same time \(t=10\) years would be \(h=f(10)+50\).

Summarizing the difference for case (b)

In the second case (b), \(h=f(t)+50\), the difference between the two sand dunes is that the second sand dune is always 50 cm higher than the first one, regardless of what the specific value of \(t\) is. #Conclusion# In conclusion, the difference between the sand dune described by \(h=f(t)\) and the sand dune described by (a) \(h=f(t+30)\) is that in case (a), the second sand dune behaves as if it is 30 years "younger" than the first sand dune, while in case (b), the second sand dune is 50 cm higher than the first sand dune for any given time \(t\).

Key concepts

Horizontal Shift

When we talk about a horizontal shift, we are referring to the movement of a function along the x-axis. In our exercise, the x-axis is represented by the time variable, \( t \). A horizontal shift occurs when we adjust the input variable by adding or subtracting a constant. - For example, if we have \( h = f(t+30) \), this represents a horizontal shift. - Adding 30 to \( t \) shifts the entire graph to the left by 30 units. This means that the second sand dune at any given time \( t \) will mirror the height of the first dune 30 years into its past.
This horizontal shift essentially means the second dune's development is seen as occurring earlier in time compared to the first dune.
The effect is as if we're viewing the second sand dune's progression 30 years behind its actual timeline, giving it a "younger" appearance.

Vertical Shift

A vertical shift involves moving the entire graph of a function up or down on the y-axis. In this context, it's about changing the output of the function, which is the height of the sand dune.- The equation \( h = f(t) + 50 \) indicates a vertical shift. Here, every height value \( h \) produced by the original function \( f(t) \) is increased by 50 cm. - This simply means the second dune's plot is straight up by 50 cm across all time values \( t \).
The practical effect of this shift is that no matter how the first sand dune changes over time, the second sand dune will always be 50 cm taller at any given time. This transformation does not affect the timing of changes but only adjusts the absolute height.

Time Variable

In this exercise, the time variable \( t \) plays a fundamental role in understanding how the sand dunes evolve. Time is often used as an independent variable in functions to model growth or change.- The time variable \( t \) tells us when the height \( h \) of the dune is being measured. - By changing \( t \) to \( t+30 \), we effectively "backdate" the second dune's growth compared to the first. This shows the power of the time variable when it directly affects how we perceive the timeline of an event or condition. Understanding the influence of the time variable allows us to manipulate and interpret functions better, offering insights into real-world changes such as erosion or growth over specific periods.

Function Notation

Function notation is a way to represent relationships between variables, typically using a formula. In our example, the function \( h = f(t) \) describes the height \( h \) of a sand dune over time \( t \).- Function notation helps in simplifying the process of referring to these complex relationships. - For instance, \( f(t+30) \) is a straightforward way to denote using the function \( f \) with the adjusted time variable \( t+30 \). In this system, we can quickly understand the relationship and changes in dependencies between height and time. This notation is remarkably flexible for expressing shifts and transformations such as horizontal and vertical shifts, enhancing the clarity of mathematical expressions in describing real-world phenomena like this erosion process.