Problem 42 - Calculus Concepts

Question

Extraterrestrial Radiation The rate of change in the rate at which the average amount of extraterrestrial radiation in Amarillo, Texas, for each month of the year is changing is proportional to the amount of extraterrestrial radiation received. The constant of proportionality is \(k=-0.212531 .\) In any given month, the expected value of radiation is \(12.5 \mathrm{~mm}\) per day. This expected value is actually obtained in March and September. (Source: Based on data from A. A. Hanson, ed. Practical Handbook of Agricultural Science, Boca Raton: CRC Press, 1990) d. How well does the model estimate the amounts of extraterrestrial radiation in March and September? a. Write a differential equation for the information given. b. In June, the amount of radiation received is approximately \(17.0 \mathrm{~mm}\) per day, and in December, the amount of radiation received is approximately 7.8 \(\mathrm{mm}\) per day. Write a particular solution for this differential equation. c. Change the particular solution into a function giving the average amount of extraterrestrial radiation in Amarillo.

Step-by-Step Solution

Answer

The model's quality depends on the accuracy of its predictions for March and September, which should approximate 12.5 mm/day each.

Step 1: Understand the Problem

We are given that the rate of change in radiation (DR ) is proportional to the amount of radiation (R). The proportionality constant is \(k = -0.212531\). We need to write a differential equation describing this relationship.

Step 2: Write the Differential Equation

The problem states that \(\frac{dR}{dt} = kR\), where \(k = -0.212531\). Using this information, the differential equation is written as:\[ \frac{dR}{dt} = -0.212531R \]

Step 3: Solve the Differential Equation

This is a separable differential equation. Separating variables gives:\[ \frac{1}{R} \frac{dR}{dt} = -0.212531 \]Integrating both sides with respect to \(t\) gives:\[ \ln |R| = -0.212531t + C \]Solving for \(R\) yields:\[ R(t) = C'e^{-0.212531t} \] where \(C' = e^C\) is a constant.

Step 4: Use Given Data to Find Particular Solution

In March and September, the radiation is \(12.5 \mathrm{~mm/day}\), which is taken as a reference. In June, \(R=17.0 \mathrm{~mm/day}\), and in December, \(R=7.8 \mathrm{~mm/day}\). We use these data points to find \(C'\) and personalize our function.

Step 5: Implement Function At Known Points

Using the information for June (around month 6), we set:\[ R(6) = 17 = C'e^{-0.212531\times 6} \]Similarly, for December (around month 12), we set:\[ R(12) = 7.8 = C'e^{-0.212531\times 12} \]Solving these equations will allow us to find a value for \(C'\).

Step 6: Establish and Manipulate Radiation Function

Solve for \(C'\) using the above equations, then adjust the general solution to a specific one by setting \(R(6) = 17\) and \(R(12) = 7.8\). Now we can predict radiation levels using our model for any month value \(t\).

Step 7: Verify Model Predictability For March and September

Since the expected value of radiation in March and September is \(12.5\ mm/day\), it is essential to verify if the model approximates these values correctly by inserting these months' respective \(t\) values into the function and checking if \(R(t) = 12.5\).

Key Concepts

Extraterrestrial Radiation

Extraterrestrial radiation refers to the solar energy that reaches the top of Earth's atmosphere. It is an essential factor as it affects climate patterns and agricultural productivity. Variations of extraterrestrial radiation throughout the year often depend on the Earth's tilt and orbit around the Sun.
In Amarillo, Texas, this radiation varies each month. Its rate of change is an interesting subject for study, especially because it tells us about how much radiation is received during different seasons. For instance, during the equinox months of March and September, the expected extraterrestrial radiation value is constant at 12.5 mm/day. This constant reading at certain times of the year allows for the validation and adjustment of models concerned with predicting radiation amounts.
By understanding the pattern of extraterrestrial radiation, scientists and environmentalists can better understand weather patterns and potentially predict agricultural yields.

Proportionality Constant

The concept of a proportionality constant is central to the differential equation at hand. It is a mathematical constant denoted by \(k\) that represents the relationship between two proportional quantities. In simpler terms, if one quantity changes, the other changes in a manner that is consistent because of the proportionality constant.
In the given exercise, the rate of change of extraterrestrial radiation is proportional to the radiation present, and the constant of proportionality is \(k = -0.212531\). This signifies that for every unit increase in time, the extraterrestrial radiation decreases (due to the negative sign) at a factor determined by \(k\).

• The effect of \(k\) being negative suggests a decrease over time.
• Appropriate manipulation of \(k\) is crucial in accurately modeling natural phenomena like solar radiation.

This constant helps in forming a reliable prediction model, by setting a fixed rate of change according to which the radiation changes.

Separable Differential Equation

A separable differential equation is a type of differential equation in which the variables can be separated on opposite sides of the equation. This type of equation allows us to integrate each side independently.
In the context of the radiation problem, the differential equation \( \frac{dR}{dt} = -0.212531R \) is separable. To solve it, we can rearrange terms to isolate \(R\) on one side and \(t\) on the other. This results in the following separation:
\[\frac{1}{R} \frac{dR}{dt} = -0.212531\]
Integrating both sides gives us a way to solve for \(R\) in terms of \(t\). The process is as follows:

• Integrate \( \frac{1}{R} dR \) to obtain \( \ln |R| \).
• Integrate the constant on the right to get \( -0.212531t + C \).

Separable differential equations are particularly helpful as they simplify complex relationships, making them manageable and solvable through integration.

Particular Solution

A particular solution to a differential equation provides a specific solution that satisfies original conditions or measurements. This type of solution is often used when there are specific data points or boundary conditions.
For the extraterrestrial radiation problem, given data points in June and December allow us to find a particular function for different months. In June, the radiation is \(17.0\) mm/day, and in December, it is \(7.8\) mm/day.
By plugging these values into the general solution, \( R(t) = C'e^{-0.212531t} \), we can solve for the constant \(C'\).

• In June, set \( R(6) = 17 = C'e^{-0.212531\times 6} \).
• In December, set \( R(12) = 7.8 = C'e^{-0.212531\times 12} \).

Solving these equations provides a unique value for \(C'\), and the particular solution tailored specifically for June and December is critical for precise predictions. It helps in confirming the model's accuracy for other times during the year by ensuring it aligns with observed data points.