Problem 17 - 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 in 2008 .

Step-by-Step Solution

Answer

The expression is \(S = f(0)\).

Step 1: Understand the Problem

We need to find the average annual sea level in Aberdeen in the year 2008 using the function \(S = f(t)\). Here, \(t\) represents the number of years before 2008.

Step 2: Identify the Given Year

Since we are looking for the sea level in 2008, and \(t\) is defined as the number of years before 2008, we need to set \(t = 0\) for the year 2008.

Step 3: Substitute Into the Expression

To find the average annual sea level in 2008, substitute \(t = 0\) into the function \(S = f(t)\). Thus, we calculate \(S = f(0)\).

Step 4: Mathematical Expression

The mathematical expression representing the average annual sea level in 2008 is \(S = f(0)\). This means you evaluate the function \(f\) at \(t = 0\) to get the sea level.

Key Concepts

Understanding Average Annual Sea Level

When we talk about average annual sea level, we refer to the height of the sea at a specific location averaged over a year. This is measured to help scientists and researchers understand trends in sea level changes over time. In our problem, we're figuring out the average annual sea level in Aberdeen, Scotland.
Aberdeen's sea level is influenced by various factors such as global warming, melting ice caps, and geological activities. These factors combined make sea levels rise or fall over years. Calculating an average helps smooth out short-term fluctuations.
Sea level data is essential for understanding climate impacts, planning coastal defenses, and predicting future changes in the environment. By using the function notation, we can model these changes over time, providing a clear mathematical approach to a complex environmental issue.

Crafting a Mathematical Expression

A mathematical expression is a statement that uses numbers, variables, and operations to represent a particular idea or relationship. In our context, the function notation \(S = f(t)\) is used to model the average annual sea level as a function of time, \(t\). This tells us that for each value of \(t\), there is a corresponding sea level \(S\).
Function notation is powerful because it provides a way to express complex relationships in a simplified and standardized form. For example, when we wrote \(S = f(0)\), we're indicating that we want the sea level when \(t\) equals 0, which corresponds to the year 2008 in the problem.
By setting up mathematical expressions, we can easily manipulate and evaluate functions for different time periods or situations, aiding in both predictions and interpretations of real-world phenomena.

Substitution in Functions

Substitution in functions is a method used to find specific values by replacing a variable with a number or another expression. This process is central in solving the problem with the average annual sea level function in Aberdeen.
When we substitute, we're effectively "plugging in" the number for \(t\) into the function \(S = f(t)\) to find the sea level for a specific year. In the example given, substituting \(t = 0\) was key since it defined the sea level for the year 2008.
Here are some steps for substitution:

• Identify the variable to replace and the value to use.
• Substitute the value into the function.
• Simplify the function if necessary to find the result.\(S = f(0)\) was simplified because \(t = 0\) meant we directly evaluated the expression for 2008.

Understanding substitution helps us analyze and interpret the real-world data expressed through functions efficiently.