Problem 54 - Applied Calculus

Question

During the 1970 s and 1980 s, the buildup of chlorofluorocarbons (CFCs) created a hole in the ozone layer over Antarctica. After the 1987 Montreal Protocol, an agreement to phase out CFC production, the ozone hole has shrunk. The ODGI (ozone depleting gas index) shows the level of CFCs present. \(^{21}\) Let \(O(t)\) be the ODGI for Antarctica in year \(t ;\) then \(O(2000)=95\) and \(O^{\prime}(2000)=-1.25 .\) Assuming that the ODGI decreases at a constant rate, estimate when the ozone hole will have recovered, which occurs when \(\mathrm{ODGI}=0\)

Step-by-Step Solution

Answer

The ozone hole is expected to completely recover by the year 2076.

Step 1: Introduction of Linear Model

Since the rate of change of the ODGI is constant, we can model the ODGI with a linear function. Let's let the function be defined as \( O(t) = mt + b \), where \( m \) is the rate of change and \( b \) is the y-intercept.

Step 2: Identify Known Values

We know that \( O(2000) = 95 \), and the rate of change \( O'(2000) = -1.25 \). Therefore, \( m = -1.25 \) and for \( t = 2000 \), \( O(t) = 95 \).

Step 3: Substitute and Solve for Intercept

Substitute the known values into the linear equation. Plug in \( t = 2000 \), \( O(2000)=95 \), and \( m = -1.25 \) into \( O(t) = mt + b \):\[95 = -1.25(2000) + b\]Now, calculate \( b \).

Step 4: Calculate Intercept

Solve for \( b \):\[95 = -1.25 \times 2000 + b \]\[95 = -2500 + b \]\[b = 95 + 2500 = 2595\]

Step 5: Write the Linear Function for ODGI

Now that we have both \( m \) and \( b \), we can write the linear function for ODGI as:\[O(t) = -1.25t + 2595\]

Step 6: Solve for the Year of Recovery

The ozone hole will have recovered when \(O(t) = 0\). Set the equation \( -1.25t + 2595 = 0 \) and solve for \(t\).

Step 7: Calculate Year of Recovery

Solve the equation:\[-1.25t + 2595 = 0 \]\[1.25t = 2595 \]\[t = \frac{2595}{1.25} = 2076\]Thus, the ozone hole is expected to have recovered by the year 2076.

Key Concepts

Ozone Layer Recovery

The ozone layer is an essential component of the Earth’s atmosphere, protecting life on Earth by absorbing most of the sun’s harmful ultraviolet radiation. The depletion of this layer, particularly over Antarctica, became a significant environmental issue during the 1970s and 1980s. A marked increase in chlorofluorocarbons (CFCs) created a hole in the ozone, leading to an urgent need for global action.

Efforts to facilitate ozone layer recovery were set into motion with international agreements like the 1987 Montreal Protocol. This treaty was crucial as it aimed to phase out the production of ozone-depleting substances. Thanks to these coordinated global efforts, the depletion has slowed, and the hole in the ozone layer has begun to shrink.

Recovery of the ozone layer is measured by the Ozone Depleting Gas Index (ODGI), which tracks the presence of chlorine and bromine compounds in the atmosphere. As science progresses, experts use models to predict when the ozone layer might fully recover. If the current trends continue, projections indicate that pre-1980s ozone levels might be restored by around the mid-21st century, providing a hopeful outlook for environmental recovery.

Chlorofluorocarbons (CFCs)

Chlorofluorocarbons, commonly known as CFCs, are chemical compounds initially developed in the 1930s for use in refrigeration, air conditioning, and aerosol propellants. These substances were popular due to their stability and non-flammable nature. However, it was later discovered that upon release into the atmosphere, CFCs contribute significantly to ozone layer depletion.

When CFCs reach the upper atmosphere, they are broken down by ultraviolet radiation, releasing chlorine atoms. These atoms participate in chemical reactions that destroy ozone molecules. This chain of reactions leads to persistent thinning of the ozone layer, particularly noticeable over polar regions such as Antarctica.

The 1987 Montreal Protocol served as a turning point by restricting the production and emission of CFCs. As regulations tightened, global CFC emissions have dramatically decreased, allowing natural atmospheric processes to gradually heal the ozone layer. The decline in CFC levels is a key factor in the observed decrease in the size of the ozone hole, highlighting the protocol's success in addressing this environmental challenge.

Constant Rate of Change

In mathematics, the concept of a constant rate of change is often used in linear modeling. It refers to a situation where a variable changes by the same amount over equal intervals of time. For the ozone layer recovery exercise, the Ozone Depleting Gas Index (ODGI) decreases at a constant rate, signified by the derivative of the function.

In our context, the rate of change at the year 2000 is given as \( \frac{dO}{dt} = -1.25 \), indicating that the ODGI decreases by 1.25 units every year. This linear decrease allows us to create a predictive model using the function \( O(t) = mt + b \), where \( m = -1.25 \) is the constant rate of change and \( b = 2595 \) is derived from the conditions known at the specific point in time.

Linear models with constant rates of change are valuable tools in estimating future values based on present and past data. In this case, solving \( O(t) = 0 \) gives the year when ODGI reaches zero, marking the theoretical full recovery of the ozone layer. Because the rate of decrease does not change over time, this simplification ensures clarity and ease of prediction in real-world applications.