Problem 88 - Elementary and Intermediate Algebra

Question

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

The ozone layer in 2015 is approximately 291.87 Dobson units thick.

Step 1: Understand the problem

We have a function \(A(t) = 300 e^{-0.0011t}\) that models the ozone layer's thickness in Dobson units over time. We need to determine the thickness in the year 2015, which is 25 years after 1990.

Step 2: Set up the problem

The problem asks us to find \(A(25)\) as \(t\) represents the number of years after 1990. So, in this case, \(t = 2015 - 1990 = 25\).

Step 3: Substitute the value of t into the function

We substitute \(t = 25\) into the function to find \(A(t)\). So we calculate \(A(25) = 300 e^{-0.0011 \times 25}\).

Step 4: Calculate the exponent

Find the value of the exponent: \(-0.0011 \times 25 = -0.0275\).

Step 5: Evaluate the exponential function

Calculate \(e^{-0.0275}\) using a calculator, which gives us approximately 0.9729.

Step 6: Calculate the thickness in 2015

Now substitute this back into the expression for \(A(25)\): \(A(25) = 300 \times 0.9729\).

Step 7: Compute the final answer

Perform the multiplication: \(300 \times 0.9729 \approx 291.87\). Therefore, the ozone layer is approximately 291.87 Dobson units thick in 2015.

Key Concepts

Mathematical Modeling

Mathematical modeling is a crucial tool in understanding and predicting natural phenomena. When we model something mathematically, we use functions or equations to represent real-world scenarios. It helps us make predictions and solve problems without the need for physical experiments, which can sometimes be impractical or impossible. In this context, the function \(A(t) = 300 e^{-0.0011 t}\) models the thickness of the ozone layer over a particular city in Dobson units across time.

• In the function, \(A(t)\) represents the thickness of the ozone layer at any given time \(t\).
• The constant \(300\) corresponds to the initial thickness in Dobson units in the year 1990.
• The exponent \(-0.0011t\) indicates a decrease rate, showing how the ozone layer's thickness changes over time.

This model uses an exponential decay formula, depicted by the negative exponent, meaning that over time the ozone layer's thickness reduces, at a rate gently decreasing each year. Understanding this model helps us predict how changes might affect the ozone layer in the future.

Ozone Layer

The ozone layer is a crucial component of Earth's atmosphere. It is primarily located in the lower portion of the stratosphere and contains high concentrations of ozone (\(O_3\)). This gas absorbs the majority of the sun's harmful ultraviolet (UV) radiation.

• Its main function is to shield the planet's surface from excessive UV radiation which can cause skin cancer and cataracts in humans and damage other living forms on Earth.
• Human activities, particularly the release of chlorofluorocarbons (CFCs), have led to depletion of the ozone layer.
• Ozone layer thinning equates to greater exposure to UV rays, affirming the importance of measuring its thickness accurately.

By using mathematical models to monitor the ozone layer's status, scientists can better understand the environmental factors affecting it and take steps to protect this important planetary shield.

Dobson Units

Dobson units (DU) are the standard measure for the thickness of the ozone layer in the atmosphere. Named after G. M. B. Dobson, a pioneer in the field of ozone research, this unit allows scientists to measure the total amount of ozone directly overhead, from the ground to the edge of space.

• One Dobson unit is equivalent to a 0.01 millimeter thickness of ozone under standard temperature and pressure conditions if it were compressed into a single layer.
• Typical ozone layers over the planet range between 200 to 500 Dobson units, with variations based on geographical and environmental factors.
• As exhaustively modeled in the given exercise, understanding changes in thickness in Dobson units helps assess the protection level of the ozone over specific areas, while providing insight into global environmental health trends.

Using Dobson units offers a straightforward method for comparing changes over time, aiding in global efforts to monitor and hopefully mitigate ozone depletion issues.