Problem 88 - Intermediate Algebra

Question

Ozone Concentrations. A Dobson unit is the most basic measure used in ozone research. Roughly 300 Dobson units are equivalent to the height of 2 pennies stacked on top of each other. Suppose the ozone layer thickness (in Dobsons) over a certain city is modeled by the function \(A(t)=300 e^{-0.0011 t}\) where \(t\) is the number of years after \(1990 .\) Estimate how thick the ozone layer will be in 2015 .

Step-by-Step Solution

Answer

Approximately 292 Dobson units.

Step 1: Understand the Problem

We want to estimate the thickness of the ozone layer in the year 2015. The thickness is modeled by the function \( A(t) = 300 e^{-0.0011 t} \), where \( t \) is the number of years after 1990.

Step 2: Identify the Time Variable

To find the correct \( t \), subtract 1990 from 2015. This gives \( t = 2015 - 1990 = 25 \).

Step 3: Substitute the Time Variable

Insert \( t = 25 \) into the ozone concentration model: \( A(25) = 300 e^{-0.0011 \, \times \, 25} \).

Step 4: Simplify the Exponent

Calculate the exponent: \( -0.0011 \times 25 = -0.0275 \).

Step 5: Calculate the Exponential Part

Evaluate \( e^{-0.0275} \). This is approximately \( 0.9729 \).

Step 6: Calculate the Thickness

Now, compute \( A(25) = 300 \times 0.9729 = 291.87 \).

Step 7: Conclusion

The estimated thickness of the ozone layer in 2015 is approximately \( 291.87 \) Dobson units.

Key Concepts

Ozone Layer Thickness Modeling

Modeling the thickness of the ozone layer is crucial for understanding how it changes over time due to various factors like pollution and natural processes. In our example, the thickness of the ozone layer is modeled using an exponential function. This function helps predict how thin or thick the layer will be in future years. Using models allows scientists and researchers to assess and plan for environmental changes.

An exponential decay model is utilized because it reflects how the ozone layer reduces over time, primarily due to human impact and natural degradation. The model in the exercise uses the function \(A(t) = 300 e^{-0.0011 t}\), where \(t\) represents the number of years since 1990. This function predicts the thickness by decreasing exponentially, showcasing a common pattern found in decay processes of natural phenomena.

Such models are vital as they provide an estimate to which decision-makers on environmental policies can refer to in order to implement measures that can possibly reverse or mitigate the impact of ozone layer decay.

Dobson Units

Dobson units (D.U.) are the standard measurement used in monitoring the thickness of the ozone layer. Named after G.M.B. Dobson, these units describe the amount of ozone in a column of the atmosphere.

• One Dobson unit represents a layer of ozone 0.01 millimeters thick at standard temperature and pressure.
• A typical ozone layer measures about 300 Dobson units, equating to a 3 millimeter thick layer of ozone around the Earth.
• Dobson units provide a straightforward method to gauge changes in ozone concentration and are useful for comparison across different regions and time frames.

In the exercise, measuring the thickness in Dobson units allows for a clear understanding of how significant the thickness of the ozone layer is compared to expected standard values. This unit helps convey whether the ozone layer is in a healthy state or if it requires attention due to depletion.

Exponential Functions

Understanding exponential functions is key to modeling and predicting phenomena like ozone layer depletion. An exponential function is a mathematical expression of the form \(f(x) = ab^x\), where \(b\) is the base of the exponential, and \(a\) is a constant multiplier.

In the context of the exercise, the function \(A(t) = 300 e^{-0.0011 t}\) is used. Here's what each part signifies:

• \(300\) is the initial thickness, representing the expected thickness of the ozone layer at time zero (1990).
• \(e\) is the base of the natural logarithm, roughly equal to 2.718, which commonly appears in continuous growth or decay processes.
• \(-0.0011\) is the decay rate, indicating that the thickness is decreasing over time as a function of \(t\).

Exponential decay functions are effective for modeling situations where quantities reduce gradually over time, such as radioactive decay, population decline, or our current example—ozone layer thickness. Becoming familiar with these functions and their properties can empower you to analyze and predict real-world processes more effectively.