Problem 60 - A Graphical Approach to College Algebra

Question

Tree Growth The height of a tree in feet after \(x\) years is modeled by $$f(x)=\frac{50}{1+47.5 e^{-0.22 x}}$$ (a) Make a table for \(f\) starting at \(x=10\) and incrementing by \(10 .\) What seems to be the maximum height? (b) Graph \(f\) and identify the horizontal asymptote. Explain its significance. (c) After how long was the tree 30 feet tall?

Step-by-Step Solution

Answer

(a) Max height approaches 50 feet; (b) Horizontal asymptote is \(y=50\); (c) Tree is 30 feet tall at 21.6 years.

Step 1: Calculate Tree Heights

To create a table for the function \(f(x)=\frac{50}{1+47.5 e^{-0.22 x}}\), we will calculate the height for different values of \(x\). We start with \(x=10\) and increment by 10 up to a chosen endpoint, say \(x=50\).- \(f(10)=\frac{50}{1+47.5 e^{-0.22 \cdot 10}}\approx 13.4\)- \(f(20)=\frac{50}{1+47.5 e^{-0.22 \cdot 20}}\approx 24.1\)- \(f(30)=\frac{50}{1+47.5 e^{-0.22 \cdot 30}}\approx 34.0\)- \(f(40)=\frac{50}{1+47.5 e^{-0.22 \cdot 40}}\approx 42.1\)- \(f(50)=\frac{50}{1+47.5 e^{-0.22 \cdot 50}}\approx 46.5\)From this table it seems the maximum height approaches 50 feet.

Step 2: Graph the Function

To graph \(f(x)=\frac{50}{1+47.5 e^{-0.22 x}}\), use graphing software or tools like Desmos. Plot the points found in the table. The curve will initially rise steeply and then level off as \(x\) increases.The horizontal asymptote is \(y=50\), indicating that the tree will approach but never exceed 50 feet in height. This asymptote represents the maximum biological height the tree can achieve under these growth conditions.

Step 3: Solve for When Tree is 30 Feet Tall

To find when the tree reaches 30 feet, set \(f(x)=30\) and solve for \(x\).\[30 = \frac{50}{1+47.5 e^{-0.22 x}}\]Cross multiply to solve:\[30(1+47.5 e^{-0.22 x}) = 50\]\[30 + 1425 e^{-0.22 x} = 50\]\[1425 e^{-0.22 x} = 20\]\[e^{-0.22 x} = \frac{20}{1425}\]Take the natural logarithm:\[-0.22x = \ln\left(\frac{20}{1425}\right)\]\[x \approx \frac{\ln\left(\frac{20}{1425}\right)}{-0.22} \approx 21.6\]Therefore, the tree was approximately 30 feet tall after 21.6 years.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions used to describe scenarios where quantities grow or decay at rates proportional to their current value. They can take various forms, such as growth, where the quantity increases over time, or decay, where the quantity decreases.
In the tree growth model, the function given by \[f(x) = \frac{50}{1 + 47.5 e^{-0.22 x}}\]describes the exponential growth of a tree's height over time. Here, the rate of growth decreases as time increases, indicating a process known as exponential decay. This function features an exponential term \(e^{-0.22 x}\), where \(e\) is the base of natural logarithms, and \(x\) represents time.

• The negative exponent suggests a diminishing growth rate, which is common in constrained biological growth.
• Over time, this will make the tree's height approach a certain value but never reach it fully.

This is because exponential functions are often used in modeling natural growth processes due to their ability to describe realistic growth behaviors.

Horizontal Asymptotes

In the context of functions and graphs, a horizontal asymptote signifies a value that a function approaches but never actually reaches. As the independent variable \(x\) trends towards positive or negative infinity, the function's value will get closer and closer to the asymptote.
For the tree height function \(f(x) = \frac{50}{1 + 47.5 e^{-0.22 x}}\), the horizontal asymptote is \(y=50\). This means the tree's maximum possible height is 50 feet, considering the environmental and genetic limits.

• This asymptote represents a biological constraint in the model, indicating the maximum height the tree can asymptotically reach.
• Graphically, this means the curve will flatten out and run parallel to the line \(y = 50\) as \(x\) approaches infinity.

Thus, understanding horizontal asymptotes is crucial when interpreting real-world growth models like this one, as they provide insights into long-term behaviors.

Table of Values

A table of values is an essential tool for understanding and interpreting functions. It provides specific outputs (or \(y\)-values) for various inputs (or \(x\)-values), allowing students to observe trends and changes directly. To construct a table for the tree growth function, evaluate the function at different \(x\)-values and record the corresponding \(f(x)\).

• At \(x=10\), the height is approximately 13.4 feet.
• At \(x=20\), the height is approximately 24.1 feet.
• At \(x=30\), the height is approximately 34.0 feet.
• At \(x=40\), the height is approximately 42.1 feet.
• At \(x=50\), the height is approximately 46.5 feet.

The table serves as a bridge between abstract mathematical functions and tangible real-world implications, helping to visualize the growth behavior. From these values, it becomes visible that the growth slows down as time progresses, approaching a maximum height of 50 feet.

Graphing Functions

Graphing functions allows us to visualize the behavior of functions over different intervals. When graphing the tree growth model, plot the computed values from the table on a coordinate system with \(x\) as the time and \(y\) as the height in feet. Connect these points smoothly to see the overall growth pattern.

• The initial steepness of the curve indicates rapid growth during the early years.
• As the curve approaches the horizontal asymptote, it flattens, showing slower growth.

Visually, you'll notice a logistic-like curve, which is typical of biological growth processes constrained by environmental factors. When the plot levels off near the asymptotic value, it signifies that the tree has approached its maximum biological height. Graphing provides an intuitive understanding, bridging the gap between equations and natural phenomena. It turns numbers and formulas into understandable and relatable visuals, which are vital in learning and communicating mathematics effectively.