Problem 32 - Calculus

Question

The function \(S=f(t)\) gives the average annual sea level, \(S,\) in meters, in Aberdeen, Scotland, \(^{1}\) as a function of \(t\), the number of years before \(2008 .\) Write a mathematical expression that represents the given statement. In 1983 the average annual sea level in Aberdeen was 7.019 meters.

Step-by-Step Solution

Answer

When \( t = 25 \), the sea level was 7.019 meters; thus, \( f(25) = 7.019 \).

Step 1: Understanding the Problem

We need to find a mathematical expression for the function \( S = f(t) \), which defines the average annual sea level \( S \) as a function of \( t \), the number of years before 2008. The information given tells us that in 1983, the sea level was 7.019 meters.

Step 2: Determine Value of \( t \)

Since \( t \) is the number of years before 2008, we need to calculate \( t \) for the year 1983. This is done by subtracting 1983 from 2008: \( t = 2008 - 1983 \).

Step 3: Calculate \( t \)

Perform the subtraction to find \( t \): \( t = 2008 - 1983 = 25 \). Therefore, in 1983, \( t = 25 \).

Step 4: Define the Expression

Since we have determined that \( t = 25 \) for the year 1983, we can now write the expression for 1983's sea level as \( f(25) = 7.019 \). This expression represents that when \( t = 25 \), the sea level \( S \) was 7.019 meters.

Key Concepts

Sea Level

Understanding sea level is crucial for analyzing environmental and geographical changes over time. Sea level represents the height at which the sea's surface stands relative to a baseline, often determined by the land. In our exercise, the sea level in Aberdeen, Scotland, is characterized as a function of time, given in years.

• The function, denoted as \( S = f(t) \), describes how sea level varies over time.
• The "baseline" year in our problem is 2008.
• Variations in sea level are influenced by factors such as climate change, melting ice caps, and tectonic activities.

Tracking sea levels is vital for predicting flooding, determining coastal policies, and understanding climate impacts. For Aberdeen, knowing past sea levels like the one in 1983 helps in forecasting future changes.

Mathematical Expression

A mathematical expression is a way to represent values and calculations using numbers and symbols. In this context, it is essential to model the sea level as a function of time.

• The function \( S = f(t) \) maps the number of years before 2008 to the sea level values.
• To create the expression for 1983's sea level, we first determine how many years before 2008 the year 1983 was; here, \( t = 25 \).
• The expression \( f(25) = 7.019 \) signifies that 25 years before 2008, in 1983, the sea level measurement was 7.019 meters.

These expressions allow us to represent historical data clearly and can be used to further analyze trends and changes in sea levels over time.

Average Annual

Average annual values are crucial statistics in many scientific and economic analyses because they smooth out short-term fluctuations and show long-term trends.

• "Average annual sea level" refers to the mean sea level recorded over a year.
• In our example, the average sea level in Aberdeen for a specific year, 1983, was 7.019 meters.
• This figure represents a single point in the continuous record of sea level measurements, used to track changes and predict future levels.

Using average annual data helps to compensate for seasonal and anomalous variations, providing a clearer picture of gradual changes. Such analyses are invaluable for planning and adaptation strategies against climate changes.