Question

Lumber Use The table shows the amounts of lumber used for residential upkeep and improvements (in billions of board-feet per year) for the years 1997 through \(2005 .\) (Source: U.S. Forest Service) \begin{tabular}{|l|c|c|c|c|c|} \hline Year & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline Amount & \(15.1\) & \(14.7\) & \(15.1\) & \(16.4\) & \(17.0\) \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|c|} \hline Year & 2002 & 2003 & 2004 & 2005 \\ \hline Amount & \(17.8\) & \(18.3\) & \(20.0\) & \(20.6\) \\ \hline \end{tabular} (a) Use Simpson's Rule to estimate the average number of board-feet (in billions) used per year over the time period. (b) A model for the data is $$ L=6.613+0.93 t+2095.7 e^{-t}, \quad 7 \leq t \leq 15 $$ where \(L\) is the amount of lumber used and \(t\) is the year, with \(t=7\) corresponding to 1997 . Use integration to find the average number of board- feet (in billions) used per year over the time period. (c) Compare the results of parts (a) and (b).

Short answer

The Simpson's Rule estimate and the model estimate should provide similar values for the average lumber used per year if the given model accurately represents the data. However, there may be some differences because Simpson's Rule is an approximation method and the model might not perfectly fit the actual data.

Step-by-step solution

Apply Simpson's Rule

Simpson's Rule, an approximation for the definite integral of a function, will be used to estimate the average amount of lumber used. First, find the interval width \( h = (b - a) / n = (15 - 7) / 8 = 1 \). The 'a' and 'b' are the lower and upper limits, while 'n' is the number of intervals (the total years given). Then apply Simpson's rule: \( \int_{a}^{b} f(x) dx \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4)... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)] \). Denote the amounts of lumber used each year as f(x), and plug in the values to find out the estimated average.

Compute average using given model

The given model is:\[ L=6.613+0.93t+2095.7e^{-t} \] For \(7 \leq t \leq 15\). Calculate the definite integral of this function over the interval [7, 15]. This requires application of the fundamental theorem of calculus and the laws of integral calculus. Further, calculate the average by dividing the definite integral by the range of the interval (15 - 7).

Compare the results

Finally, compare the results obtained from Simpson's Rule and the given model. Discuss whether the two methods give similar results and the possible reasons for any discrepancies.

Key concepts

Simpson's Rule

Understanding Simpson's Rule involves grasping both its purpose and its mechanics. Designed for numerical integration, Simpson's Rule gives us an approximation method for computing the definite integral of a function when exact integration is challenging. It's especially useful when dealing with data points from real-world measurements, such as the lumber usage in the given problem, or when the function itself is known only through such discrete points.

To apply Simpson's Rule, we divide the interval of integration into an even number of equally spaced subintervals. We then estimate the area under the curve using a combination of parabolas that pass through the function's values at these subinterval points. In mathematical terms, if we have a function f(x) that we want to integrate from a to b over n subintervals (where n is an even number), the rule is expressed as:$$\int_{a}^{b} f(x) dx \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 4f(x_{n-1}) + f(x_n)]$$Here, h represents the width of each subinterval ((b - a) / n), and x_0, x_1, ..., x_n stand for the subinterval endpoints and midpoints. The pattern of coefficients (1, 4, 2, ...) reflects the weight given to each function value in the formula, with alternating weights optimized for the parabolic segments used in the approximation.

Definite Integral

Central to the concept of integral calculus is the definite integral, which quantifies the accumulated quantity, such as area under a curve, within a given interval. It's represented as:$$\int_{a}^{b} f(x) dx$$ where a and b are the lower and upper limits of integration, respectively, and f(x) is the function being integrated. Calculating the definite integral allows us to understand how a function accumulates value over an interval. In contexts like the lumber use exercise, it enables us to calculate the total usage of lumber over several years.

Using the fundamental theorem of calculus, we can compute definite integrals when we have the antiderivative of the function. However, when the function is difficult to integrate symbolically or is represented by data points, we often turn to numerical methods like Simpson's Rule to estimate the integral. When applying these methods, it's essential to remember that we are approximating, although often quite accurately, the exact value of the integral. The precision of such methods depends on factors such as the nature of the function and the number of subintervals used.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario in order to analyze it and make predictions. In our lumber use exercise, a mathematical model has been provided to represent the amount of lumber used over time. This model takes the form of a function that considers both linear and exponential components, shown as:$$L = 6.613 + 0.93t + 2095.7 e^{-t}$$In this model, L is the lumber used, and t represents time, with a specific year corresponding to t=7. By integrating this function over a given interval, we can determine the total amount of lumber used during that period. Furthermore, the average lumber usage per year can be found by dividing the result of this integration by the number of years within the interval.

Mathematical models like the one in our exercise are instrumental in predicting trends and optimizing resource management in industries such as forestry. However, it's important to recognize that models are simplifications of reality. They rely on assumptions and may not account for all variables affecting the real system. Regularly validating and adjusting the model with new data is crucial to ensure its accuracy and relevance.