Problem 42 - Brief Applied Calculus

Question

Global Temperatures The burning of oil, coal, and other fossil fuels generates "greenhouse gasses" that trap heat and raise global temperatures. Although predictions depend upon assumptions of countermeasures, one study predicts an increase in global temperature (above the 2000 level) of \(T(t)=0.25 t^{1.4}\) degrees Fahrenheit, where \(t\) is the number of decades since 2000 (so, for example, \(t=2\) means the year 2020). Find \(T(10), T^{\prime}(10)\), and \(T^{\prime \prime}(10)\), and interpret your answers. [Note: Rising temperatures could adversely affect weather patterns and crop yields in many areas.]

Step-by-Step Solution

Answer

The predicted temperature increase by 2100 is 6.28°F, with temperatures rising at 0.8785°F/decade and the rate of increase accelerating.

Step 1: Identify the Function

The function given is \( T(t) = 0.25 t^{1.4} \). This function represents the increase in global temperature above the 2000 level, where \( t \) is the number of decades since 2000.

Step 2: Compute T(10)

Substitute \( t = 10 \) into the function \( T(t) = 0.25 t^{1.4} \) to find \( T(10) \).\[ T(10) = 0.25 imes 10^{1.4} \] First, calculate \( 10^{1.4} \). \[ 10^{1.4} \ \approx 25.12 \]Multiply by 0.25: \[ T(10) = 0.25 imes 25.12 \ \approx 6.28 \] Thus, \( T(10) = 6.28 \) degrees Fahrenheit.

Step 3: Differentiate to Find T'(t)

Find the first derivative of \( T(t) = 0.25 t^{1.4} \) using the power rule. The derivative of \( t^n \) is \( n t^{n-1} \).\[ T'(t) = 0.25 imes 1.4 imes t^{0.4} \]Simplify: \[ T'(t) = 0.35 t^{0.4} \]

Step 4: Compute T'(10)

Substitute \( t = 10 \) into \( T'(t) = 0.35 t^{0.4} \) to find \( T'(10) \).\[ T'(10) = 0.35 imes 10^{0.4} \] First, calculate \( 10^{0.4} \). \[ 10^{0.4} \ \approx 2.51 \]Multiply by 0.35 to get:\[ T'(10) = 0.35 imes 2.51 \ \approx 0.8785 \] Thus, \( T'(10) = 0.8785 \) degrees Fahrenheit per decade.

Step 5: Differentiate Again for T''(t)

Differentiate \( T'(t) = 0.35 t^{0.4} \) to find the second derivative. Again, use the power rule.\[ T''(t) = 0.35 imes 0.4 imes t^{-0.6} \]Simplify: \[ T''(t) = 0.14 t^{-0.6} \]

Step 6: Compute T''(10)

Substitute \( t = 10 \) into \( T''(t) = 0.14 t^{-0.6} \) to find \( T''(10) \).\[ T''(10) = 0.14 imes 10^{-0.6} \] Calculate \( 10^{-0.6} \): \[ 10^{-0.6} \ \approx 0.251 \]Multiply by 0.14:\[ T''(10) = 0.14 imes 0.251 \ \approx 0.0351 \] Thus, \( T''(10) = 0.0351 \) degrees Fahrenheit per decade squared.

Step 7: Interpret the Results

The value \( T(10) = 6.28 \) represents the predicted increase in global temperature by the year 2100. The derivative \( T'(10) \approx 0.8785 \) suggests that temperatures are increasing at a rate of approximately 0.8785 degrees per decade around the year 2100. The second derivative \( T''(10) \approx 0.0351 \) indicates that the rate of increase is itself increasing, suggesting accelerating temperature rise.

Key Concepts

Global Temperature Modeling

Global temperature modeling is an essential task in understanding climate change. Scientists use various mathematical models to predict future temperature changes based on historical data and current trends. The function given in the exercise, \( T(t) = 0.25 t^{1.4} \), models the rise in temperatures above the 2000 level, where \( t \) represents decades since 2000. This model helps to quantify how factors, such as greenhouse gas emissions, contribute to global warming.

Understanding the structure of such models is crucial in predicting potential impacts on global weather patterns and crop yields. By having a model like \( T(t) \), scientists and policymakers can strategize and implement countermeasures to mitigate negative effects. This function helps illustrate the expected growth of temperature over time, providing a quantitative look at the challenges posed by climate change.

Derivatives in Calculus

Derivatives play a vital role in calculus as they provide information about the rate at which things change. In the context of this exercise, finding the derivative of the function \( T(t) \) allows us to determine how rapidly temperatures are expected to rise over time.

The first derivative, \( T'(t) \), gives the rate of change of temperature concerning time. For the function \( T(t) = 0.25 t^{1.4} \), the first derivative is \( T'(t) = 0.35 t^{0.4} \). This tells us the approximate rate at which the temperature is rising per decade. Moreover, taking the second derivative, \( T''(t) = 0.14 t^{-0.6} \), provides insights into how this rate of change itself changes over time, indicating patterns such as acceleration or deceleration in temperature rise.

Calculating these derivatives helps not just in understanding current trends, but also in predicting future conditions, making derivatives incredibly powerful in calculus applications.

Rate of Change in Temperature

Understanding the rate of change in temperature is critical for predicting the impacts of climate change. The rate of change is essentially how fast the temperature is rising or falling over a given period. In this exercise, this is represented by the first derivative of the temperature function.

The rate of change, \( T'(10) = 0.8785 \), indicates that in year 2100, the temperature is projected to increase by roughly 0.8785 degrees Fahrenheit per decade. This measurement highlights not only the magnitude of climate change but also provides a basis for analyzing and enacting policy reforms.

Moreover, the second derivative, \( T''(10) = 0.0351 \), hints at the acceleration of this change, meaning that the rate at which temperatures are increasing is itself increasing around 2100. Acceleration implies a potential compounding effect on climate changes, making it necessary for continued study and analysis.