Problem 29 - Applied Calculus

Question

In \(2009,\) a study was done on the impact of sea-level rise in the mid- Atlantic states. \(^{26}\) Let \(a(t)\) be the depth of the sea in millimeters (mm) at a typical point on the Atlantic Coast, and let \(m(t)\) be the depth of the sea in \(\mathrm{mm}\) at a typical point on the Gulf of Mexico, with time \(t\) in years since data collection started. (a) The study reports "Sea level is rising and there is evidence that the rate is accelerating." What does this statement tell us about \(a(t)\) and \(m(t) ?\) (b) The study also reports "The Atlantic Coast and the Gulf of Mexico experience higher rates of sea-level rise ( 2 to 4 mm per year and 2 to 10 mm per year, respectively) than the current global average \((1.7 \mathrm{mm}\) per year)." What does this tell us about \(a(t)\) and \(m(t) ?\) (c) Assume the rates at which the sea level rises on the Atlantic Coast and the Gulf of Mexico are constant for a century and within the ranges given in the report. (i) What is the largest amount the sea could rise on the Atlantic Coast during a century? Your answer should be a range of values. (ii) What is the shortest amount of time in which the sea level in the Gulf of Mexico could rise 1 meter?

Step-by-Step Solution

Answer

(a) \(a(t)\) and \(m(t)\) are increasing with increasing rates. (b) \(a'(t)\) ranges from 2-4 mm/year and \(m'(t)\) from 2-10 mm/year. (c) (i) 200-400 mm rise for Atlantic Coast in a century; (ii) 100 years for Gulf of Mexico to rise 1 meter.

Step 1: Interpret the Rise and Acceleration Statement

The statement "Sea level is rising and there is evidence that the rate is accelerating" implies that both \(a(t)\) and \(m(t)\) are functions representing sea level that are increasing over time. Additionally, the acceleration implies that their second derivatives \(a''(t)\) and \(m''(t)\) are positive, indicating that the rate of rise for both the Atlantic and Gulf coasts is itself increasing.

Step 2: Analyze the Higher Rates of Sea-Level Rise

The study reports rates of 2 to 4 mm per year for the Atlantic Coast and 2 to 10 mm per year for the Gulf of Mexico, both higher than the global average of 1.7 mm per year. This tells us that the first derivatives \(a'(t)\) and \(m'(t)\), representing the rates of sea level rise, are within the specified ranges for each location.

Step 3: Calculate Maximum Sea Rise for the Atlantic Coast

Assuming a constant rate of rise for a century, the maximum rise \(R_a\) on the Atlantic Coast would be calculated by multiplying the maximum rate by 100 years: \( R_a = 4 \, \text{mm/year} \times 100 \, \text{years} = 400 \, \text{mm} \). The range for the potential sea rise on the Atlantic Coast is 200 to 400 mm over a century.

Step 4: Determine Time for 1 Meter Rise in Gulf of Mexico

Convert 1 meter to millimeters: 1 meter = 1000 millimeters. Using the maximum rate of rise for the Gulf of Mexico (10 mm/year), calculate the minimum time \(T\) for a 1 meter rise: \( T = \frac{1000 \, \text{mm}}{10 \, \text{mm/year}} = 100 \, \text{years} \). Therefore, the shortest time frame for a 1 meter rise is 100 years.

Key Concepts

Sea-Level Rise

Sea-level rise refers to the increasing height of the world's ocean surfaces. This global phenomenon results from factors like thermal expansion, where water expands as it warms, and the melting of glaciers and ice sheets. In our study, we focus on the sea levels at the Atlantic Coast and the Gulf of Mexico.

These regions are experiencing a rise in sea level that may affect coastal communities, ecosystems, and economic activities. In particular, sea-level rise can lead to increased flooding, erosion of coastlines, and saltwater intrusion into freshwater sources. Understanding how sea levels rise over time helps us develop strategies to adapt to and mitigate these impacts.

Rate of Change

The rate of change is a fundamental concept in calculus that describes how a quantity changes over time. When we talk about the rate of change in the context of sea-level rise, we are interested in how quickly the sea-level is increasing each year.

In the study, the reported rate of change for the Atlantic Coast is between 2 to 4 mm per year, while the Gulf of Mexico's rate is between 2 to 10 mm per year. These rates exceed the global average of 1.7 mm per year, indicating that these regions are more severely impacted by rising sea levels.

Understanding the rate of change helps policymakers and scientists make informed decisions about coastal management and planning.

Derivatives

Derivatives in calculus measure the rate of change of a function with respect to a variable. In our exercise, the derivative of sea level with time, represented by functions like \(a'(t)\) and \(m'(t)\), shows how fast the sea level is rising at any given moment.

Moreover, the acceleration mentioned in the study refers to the second derivative \(a''(t)\) and \(m''(t)\). A positive second derivative indicates that not only is the sea level rising, but the rate at which it rises is also increasing over time. This understanding helps highlight the urgency and dynamically growing nature of sea-level rise, requiring continuous monitoring and possibly more drastic measures to counteract the effects.

Atlantic Coast

The Atlantic Coast, stretching from the southern tip of Florida up to the icy waters of Maine, is experiencing a significant rise in sea levels. In this exercise, it is noted that the sea level here rises between 2 to 4 mm per year.

Over a century, with a consistent rate, this could lead to a range of sea level increases from 200 mm to 400 mm. Coastal cities in this region are at risk and may face challenges such as property damage from increased flooding, loss of habitat, and infrastructure stress.

To mitigate these challenges, communities along the Atlantic Coast may need to invest in flood defenses, coastal restoration projects, and revise urban planning and zoning laws to enhance resilience against sea-level rise.

Gulf of Mexico

The Gulf of Mexico is another key focus area for sea-level rise in this study. The reported rates here are notably higher, ranging from 2 to 10 mm per year.

In the worst-case scenario, the sea could rise 1 meter in about 100 years, given a maximum rise rate. This is due to both natural factors and human activity in this vibrant region known for its major ports, wetlands, and oil industries.

Increased sea-level rise could threaten coastal ecosystems, erode beaches, and impact housing and infrastructure. To cope with these changes, efforts might include significant investment in coastal defenses and adopting policies to reduce environmental impacts on this vital coastline.