Problem 93 - College Algebra and Calculus: An Applied Approach

Question

Blue Oak The blue oak tree, native to California, is known for its slow rate of growth. Fencing enclosures protect seedlings from herbivore damage and promote faster growth. The height \(H\) (in inches) of an enclosed blue oak tree can be approximated by the model \(H=0.74 t^{2}+25, \quad 0 \leq t \leq 5\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. Use the model to approximate the year in which the height of the tree was about 32 inches.

Step-by-Step Solution

Answer

The blue oak tree will be approximately 32 inches tall around the year 2001.

Step 1: Substite Given Height Into the Equation

To find the year when the height of the tree was about 32 inches, substitute \(H=32\) into the given equation: \(32=0.74t^2+25\)

Step 2: Rearrange the Equation to Solve for t

Subtract 25 from both sides to isolate the quadratic term on one side of the equation. This results in \(0.74t^2=7\). Divide by 0.74 to solve for \(t^2\). That yields \(t^2=7/0.74\).

Step 3: Solve for t

Take the square root of both sides to solve for \(t\). Keep in mind that the value of \(t\) must be a non-negative number because it represents time (a negative year doesn't make sense in this context). Therefore, discard any negative root and retain the positive root: \(t= \sqrt{7/0.74}\).

Step 4: Translate t into a Year

Since the year 2000 corresponds to \(t=0\), add the value of \(t\) found previously to 2000 to find the year when the tree is approximately 32 inches tall.

Key Concepts

Modeling Growth

Understanding the concept of modeling growth is essential in predicting future outcomes. This particular exercise uses the height of blue oak trees to illustrate growth over time. The formula given allows us to estimate the height of a tree as it grows within a controlled environment.

Why do we use models? In science, models simplify complex real-world phenomena, enabling us to make calculated predictions. In this exercise, the model is a quadratic equation, which is powerful in mapping growth that changes at varying rates.

• The equation provided is in the form of a quadratic expression: \(H = 0.74t^2 + 25\).
• "\(H\)" represents the height at any given time "\(t\)".
• The coefficients and constants, like 0.74 and 25, are derived from data that reflect the growth patterns observed in blue oak trees.

As you can see, models like these are invaluable for projecting growth and change over time. Whether in science or economics, modeling provides insight into how variables interact.

Quadratic Formula

The quadratic formula is a critical tool when working with quadratic equations. It is used to find the roots of a quadratic equation, or simply put, the values of \(t\) in our formula. The quadratic formula is:

• \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

For this exercise, even though the quadratic formula wasn't explicitly used, it's important to know it offers a method to find solutions when simple rearrangement doesn't work.

Quadratic equations describe many natural phenomena due to their parabolic shape, which can represent growth, speed, or even acceleration over time.

• In the equation \(H=0.74t^2 + 25\), "a" is 0.74, "b" is 0, and "c" is 25.

Understanding how each component affects the equation's shape is crucial for interpreting the data accurately. For problems that cannot be easily rearranged for an answer, the quadratic formula offers an important alternative solution strategy.

Problem Solving Steps

Problem-solving in mathematics, like in the exercise, typically follows a structured approach. Here's a breakdown of how to tackle a similar problem using systematic steps.
First, identify what you are looking to find. In this case, it was the year a tree reached a certain height. Apply these steps:

• Substitute: Insert the known value into the equation, resulting in \(32 = 0.74t^2 + 25\).
• Isolate the Variable: Rearrange the equation to isolate \(t\) by subtracting 25 from both sides.
• Solve the Equation: Simplify and solve for \(t\), either by traditional arithmetic means or using the quadratic formula, if necessary.

Finally, interpret the solution in the context of the question. Since \(t = 0\) corresponds to the year 2000, convert your answer from \(t\) to the appropriate year by adding your result to 2000.
Each step builds upon the previous one, helping to ensure accuracy and a solid understanding of the process.