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Chapter 11: Problem 52

The increase in carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) in the atmosphere is a major cause of global warming. Using data obtained by Charles David Keeling, professor at Scripps Institution of Oceanography, the average amount of \(\mathrm{CO}_{2}\) in the atmosphere from 1958 through 2007 is approximated by \(A(t)=0.010716 t^{2}+0.8212 t+313.4 \quad(1 \leq t \leq 50)\) where \(A(t)\) is measured in parts per million volume (ppmv) and \(t\) in years, with \(t=1\) corresponding to 1958 . Find the average rate of increase of the average amount of \(\mathrm{CO}_{2}\) in the atmosphere from 1958 through 2007 .

Short Answer

Expert verified
The average rate of increase of the average amount of CO鈧� in the atmosphere from 1958 through 2007 is approximately 1.352716 ppmv/year.

Step by step solution

01

Identify the given function and the time interval

The given function that represents the average amount of CO鈧� in the atmosphere is: \[A(t)=0.010716 t^{2}+0.8212 t+313.4\] The time interval we are considering is from \(t=1\) (1958) to \(t=50\) (2007).
02

Calculate the derivative of the given function

We will differentiate the function \(A(t)\) with respect to \(t\) to get the instantaneous rate of increase of CO鈧� in the atmosphere. The derivative can be calculated as follows: \[\frac{dA}{dt} = \frac{d}{dt}(0.010716 t^{2}+0.8212 t+313.4)\] Using the power rule and sum/difference rule for differentiation, we get: \[\frac{dA}{dt} = 2 \times 0.010716 t + 0.8212\] Now, let's rewrite the derivative function in a more simplified form: \[\frac{dA}{dt} = 0.021432t + 0.8212\]
03

Find the values of the derivative at the given time limits

Now we need to find the values of the derivative at the given time limits (1958 and 2007) i.e. at \(t=1\) and \(t=50\). For \(t=1\) (1958): \[\frac{dA}{dt} = 0.021432(1) + 0.8212 = 0.842632\] For \(t=50\) (2007): \[\frac{dA}{dt} = 0.021432(50) + 0.8212 = 1.8628\]
04

Compute the average rate of increase

Finally, to find the average rate of increase of the average amount of CO鈧� in the atmosphere from 1958 through 2007, we take the average of the values of the derivative at \(t=1\) and \(t=50\): \[\text{Average rate of increase} = \frac{0.842632 + 1.8628}{2} = \frac{2.705432}{2} = 1.352716\] So, the average rate of increase of the average amount of CO鈧� in the atmosphere from 1958 through 2007 is approximately 1.352716 ppmv/year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Environmental Mathematics
Environmental mathematics is a branch of applied mathematics that focuses on problems and questions related to the environment. This field uses mathematical tools to model, analyze, and interpret data on environmental issues, such as pollution, conservation, and climate change. In the exercise, we observe the use of a quadratic function to represent the average concentration of atmospheric CO2 over a set period, a real-world application of environmental mathematics.

Using historical data collected by scientist Charles David Keeling, the function serves to approximate the growth of CO2 concentration. This kind of mathematical modeling is crucial for understanding environmental trends and making predictions. Calculating the average rate of increase in CO2 is a prime example of how mathematics aids in grasping the scale and speed of environmental changes, ultimately informing policy decisions and scientific research.
The Role of Derivative Calculation
Derivative calculation in calculus provides information about the rate of change of a function at any given point. The derivative of a function at a particular point can be thought of as the slope of the curve of the function at that point. This concept is illustrated in the textbook example through the calculation of the derivative of the CO2 concentration function over time.

The derivative function obtained in Step 2, \[\frac{dA}{dt} = 0.021432t + 0.8212\], represents the instantaneous rate of increase of CO2 levels for any given year. It's essential to understand this calculation as it indicates how rapidly CO2 concentrations are changing each year鈥攊nformation that can be used to predict future conditions and potentially guide environmental policies.
Applied Calculus in Environmental Analysis
Applied calculus is used to solve problems in a variety of real-world contexts, extending far beyond the realm of pure mathematics. In our environmental case, applied calculus helps quantify the increase of CO2 concentration in the atmosphere. After deriving the function for the instantaneous rate of change, applied calculus techniques are used to evaluate the function at specific points in time. This approach demonstrates a practical application of calculus in analyzing and interpreting environmental data.

In the given exercise, applied calculus is used to estimate the average rate of increase in atmospheric CO2 from 1958 to 2007. By averaging the derivatives at the start and end of the time interval, we gain a comprehensive view of the change over the 50-year period. Steps 3 and 4 of the solution showcase how calculus can provide meaningful summaries of environmental trends, a testament to its value in addressing ecological issues.

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