Problem 42 - Calculus

Question

Chlorofluorocarbons (CFCs) were used as propellants in spray cans until their buildup in the atmosphere started destroying the ozone, which protects us from ultraviolet rays. since the 1987 Montreal Protocol (an agreement to curb CFCs), the CFCs in the atmosphere above the US have been reduced from a high of 1915 parts per trillion (ppt) in 2000 to 1640 ppt in \(2014 .^{21}\) The reduction has been approximately linear. Let \(C(t)\) be the concentration of CFCs in ppt in year \(t.\) (a) Find \(C(2000)\) and \(C(2014)\) (b) Estimate \(C^{\prime}(2000)\) and \(C^{\prime}(2014)\) (c) Assuming \(C(t)\) is linear, find a formula for \(C(t)\) (d) When is \(C(t)\) expected to reach 1500 ppt, the level before CFCs were introduced? (e) If you were told that in the future, \(C(t)\) would not be exactly linear, and that \(C^{\prime \prime}(t)>0,\) would your answer to part (d) be too early or too late?

Step-by-Step Solution

Answer

C(2000) = 1915, C(2014) = 1640, C'(t) = -19.64 ppt/year, C(t) ≈ 1500 ppt by late 2021, the 2021 prediction might be too early if C''(t) > 0.

Step 1: Identify C(2000) and C(2014)

We are given that in 2000, the concentration of CFCs was at its peak with 1915 ppt, so \( C(2000) = 1915 \). In 2014, the concentration reduced to 1640 ppt, therefore \( C(2014) = 1640 \).

Step 2: Estimate C'(2000) and C'(2014)

Given that the reduction is linear, the rate of change, \( C'(t) \), is constant. Therefore, \( C'(2000) = C'(2014) = \frac{C(2014) - C(2000)}{2014 - 2000} = \frac{1640 - 1915}{14} = -19.64 \) ppt per year.

Step 3: Find the Linear Formula for C(t)

The linear equation can be written as \( C(t) = mt + b \), where \( m = C'(2000) = -19.64 \). Using the point \( (2000, 1915) \), we solve for \( b \): \[1915 = -19.64 \times 2000 + b \]\[b = 1915 + 39280 = 41195 \]Thus, the equation is \( C(t) = -19.64t + 41195 \).

Step 4: Calculate When C(t) Reaches 1500 ppt

To find \( t \) when \( C(t) = 1500 \), solve the equation: \[-19.64t + 41195 = 1500\]\[-19.64t = 1500 - 41195 = -39695 \]\[t = \frac{39695}{19.64} \approx 2021.9 \]Thus, \( C(t) \) is expected to reach 1500 ppt around late 2021.

Step 5: Evaluate C'(t) Change Over Time

If \( C'(t) \) remains negative but \( C''(t) > 0 \), \( C(t) \) will decrease at a slower rate over time, meaning it would take longer to reach 1500 ppt. Hence, the previous estimate could be too early.

Key Concepts

Chlorofluorocarbons (CFCs)

Chlorofluorocarbons, commonly abbreviated as CFCs, are compounds that contain chlorine, fluorine, and carbon. For many years, CFCs were widely used as propellants in aerosol spray cans, refrigerants, and as solvents in the manufacture of foam products. However, in the late 20th century, scientists discovered that CFCs released into the atmosphere were causing significant harm to the ozone layer—a part of the Earth's stratosphere that absorbs most of the sun's harmful ultraviolet radiation.
CFCs were found to be very stable, enabling them to travel to the upper atmosphere, where ultraviolet light breaks them down, releasing chlorine atoms. These chlorine atoms, in turn, destroy ozone molecules, leading to depletion of the ozone layer.
The Montreal Protocol, signed in 1987, was a crucial international agreement aimed at phasing out the production and consumption of ozone-depleting substances, including CFCs. Since then, there has been a significant reduction in the concentration of CFCs in the atmosphere.

• This reduction demonstrates the impact of coordinated global efforts on environmental recovery.
• From 2000 to 2014, CFC levels dropped from 1915 to 1640 parts per trillion in the U.S., showing a promising linear trend towards decreasing levels.

Rate of Change

The rate of change is a fundamental concept in calculus that describes how one quantity changes with respect to another. In the context of the given exercise, the rate of change refers to how the concentration of CFCs in the atmosphere decreases over time.
For a linear function, the rate of change is constant, which is why it can be denoted using the symbol of the derivative, \(C'(t)\). This rate can be calculated by taking the difference in CFC concentrations over a time interval and dividing by the length of that time interval. The formula used here is:

• \(C'(t) = \frac{C(2014) - C(2000)}{2014 - 2000} = \frac{1640 - 1915}{14} = -19.64\) ppt per year.

This negative value indicates a decrease in CFC levels per year, confirming that the reduction is consistent and linear. Understanding the rate of change helps us to predict future behavior of CFC concentrations under current trends.

Linear Approximation

Linear approximation is a key tool in calculus used to approximate the values of a function using a linear equation. When changes are roughly constant, like the decrease of CFCs over time, a linear approximation can effectively swap a potentially complicated relationship for a simpler linear one.
The linear function representing CFC levels over time is \(C(t) = mt + b\), where the slope \(m\) equals the rate of change, and \(b\) is the y-intercept, which is calculated using the known point on the line, \((2000, 1915)\). The equation derived is:

• \(C(t) = -19.64t + 41195\)

This formula allows us to estimate the concentration of CFCs at any time \(t\), provided the rate of decrease remains linear. It gives a useful method to forecast future concentrations and is used to determine when CFC levels will return to pre-industrial levels.

Differential Calculus

Differential calculus is a branch of mathematics focused on the study of rates at which quantities change. It's instrumental when dealing with problems related to curves and slopes of functions.
In the exercise, differential calculus helps understand how the concentration of CFCs decreases over time. By considering the simple linear function that models this problem, we use differentiation to gauge the rate of CFC reduction and predict future trends.
If we analyze the function's derivative, \(C'(t)\), it represents the constant rate of decrease. If this function was instead nonlinear, with an additional derivative \(C''(t)\), indicating change of pace in rate, one would see \(C(t)\) decreasing at a slower rate over time.
Here's where differential calculus shines:

• If \(C''(t) > 0\), it implies acceleration in reduction slowdown, showing how quickly the rate of decrease is changing.
• This insight helps adjust predictions, as it means CFCs would reach lower concentrations later than earlier projected based on a simple linear model.

Understanding these differential calculus applications is vital to navigating real-world scenarios like environmental recovery and policy impact assessments.