Problem 18 - Applied Calculus

Question

The function \(S=f(t)\) gives the average annual sea level, \(S\), in meters, in Aberdeen, Scotland, \(^{5}\) as a function of \(t,\) the number of years before \(2008 .\) Write a mathematical expression that represents the given statement. The average annual sea level in Aberdeen was the same in 1865 and 1911.

Step-by-Step Solution

Answer

The equation is \( f(143) = f(97) \).

Step 1: Identify Known Values

From the problem statement, we identify that the years given, 1865 and 1911, are referenced relative to the year 2008. Therefore, we need to find how many years before 2008 these dates are. For 1865, it is 2008 - 1865 = 143 years before. For 1911, it is 2008 - 1911 = 97 years before.

Step 2: Define the Variables

Let the variable representing the years before 2008 be denoted by \( t \). Then for the year 1865, \( t = 143 \), and for the year 1911, \( t = 97 \).

Step 3: Set Up the Equation

Since the problem states the sea levels were the same for these two years, we set the function equal for these two time periods: \( f(143) = f(97) \). This equation expresses that the average annual sea level at \( t = 143 \) years before 2008 is the same as at \( t = 97 \) years before 2008.

Key Concepts

Mathematical Modeling

Mathematical modeling is a way of using mathematical equations and functions to represent real-world situations. It's like making a 'math-based recipe' to explain or predict things we observe in our daily lives. In this exercise, the sea level over time in Aberdeen, Scotland is modeled by the function \( S = f(t) \). This function helps us understand how the sea level changes depending on the number of years before 2008. Rather than directly using years like 1865 or 1911, we convert these into variables that are easier to handle in equations without any confusion about the timeline.

• By identifying crucial data, like years and sea levels, we can create models that reflect history or make future predictions.
• This approach simplifies complex data into something we can easily manipulate and further analyze.

Overall, mathematical modeling transforms physical, scientific, or economic observations into precise mathematical language.

Functions and Graphs

Functions and their graphs represent how quantities change and relate to one another. In our scenario, the function \( S = f(t) \) indicates the sea level as a function of time \( t \). Here, \( t \) represents the years before 2008, allowing you to pinpoint when certain changes occurred in history.

• The function \( f(t) \) allows us to visualize data over time, much like plotting a historical timeline graph to see changes through years.
• Graphs of functions help in identifying trends, such as periods of increased sea levels or stabilization.

For instance, by examining where \( f(t) \) meets itself, like in our example where \( f(143) = f(97) \), we see that despite being 46 years apart, the sea levels were identical in 1865 and 1911, which can further open discussions into why this might have occurred.

Real-World Applications

The real strength of applied calculus lies in its broad real-world applications, from predicting future climate behavior to informing urban planning. The given exercise highlights just one example of how calculus ideas translate into practical applications. By understanding how sea levels change over time, decision-makers can tackle important issues such as coastal flooding and its long-term management.

• Data obtained from functions like \( S = f(t) \) supports crucial infrastructural and environmental planning tasks.
• Equations that show sea levels returning to the same value over different periods might prompt studies into environmental conditions of those times.

By applying these concepts, cities like Aberdeen can prepare better for future challenges, using historical data as a foundation for future strategies.