Question

Let \(b\) represent the base diameter of a conifer tree and let \(h\) represent the height of the tree, where \(b\) is measured in centimeters and \(h\) is measured in meters. Assume the height is related to the base diameter by the function \(h=5.67+0.70 b+0.0067 b^{2}\). a. Graph the height function. b. Plot and interpret the meaning of \(\frac{d h}{d b}\).

Short answer

Answer: The height of a conifer tree increases with respect to its base diameter according to the function \(h(b) = 5.67 + 0.70 b + 0.0067 b^{2}\). The rate of change in height, represented by the derivative function \(\frac{dh}{db} = 0.70 + 0.0134 b\), shows that the height increases at a faster rate as the base diameter increases. In other words, the taller the tree, the faster it is growing in height relative to its base diameter.

Step-by-step solution

Identify the height function

The given height function is: \(h(b) = 5.67 + 0.70 b + 0.0067 b^{2}\)

Graph the height function

To graph the height function, you can use a graphing calculator or plotting software. Simply plot the function \(h(b) = 5.67 + 0.70 b + 0.0067 b^{2}\) on a graph. Set an appropriate range for the x-axis (base diameter) and y-axis (tree height) based on real-world parameters. Here's a sample graph: https://www.wolframalpha.com/input/?i=plot+5.67%2B0.70b%2B0.0067b%5E2

Compute the derivative of the height function with respect to base diameter

The derivative of the height function, \(\frac{dh}{db}\), represents how the height of the tree changes with respect to its base diameter. To find the derivative, we use the power rule: \(\frac{d}{db}(5.67 + 0.70 b + 0.0067 b^2) = 0 + 0.70 + 0.0134 b\) So, \(\frac{dh}{db} = 0.70 + 0.0134 b\)

Plot the derivative of the height function with respect to base diameter

Similar to Step 2, we will plot the derivative function using a graphing calculator or plotting software. Plot the function: \(\frac{dh}{db} = 0.70 + 0.0134 b\) Here's a sample graph: https://www.wolframalpha.com/input/?i=plot+0.70+%2B+0.0134b

Interpret the meaning of the derivative function

The plotted graph of \(\frac{dh}{db}\) shows that the rate of change of the tree's height with respect to its base diameter is positive and increases linearly with the base diameter. This means that as the tree's base diameter increases, its height increases at a faster rate. In other words, the taller the tree, the faster it is growing in height relative to its base diameter.

Key concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this exercise, the height function of a tree is given as a polynomial:

• \(h(b) = 5.67 + 0.70b + 0.0067b^2\)

This equation is a second-degree polynomial in terms of base diameter \(b\), where each term is a product of a constant and a power of \(b\). The first term, 5.67, is a constant indicating a base height of the tree. The second term, 0.70b, represents the linear contribution of the base diameter to the tree's height. The third term, 0.0067b^2, is quadratic and reflects how the growth rate changes as the base diameter increases.
Polynomial functions are essential in modeling real-world scenarios like this, as they can describe complex relationships through relatively simple calculations.

Rate of Change

The rate of change in this context refers to how the tree's height changes in response to changes in its base diameter. This can be calculated using the derivative of the height function, \(\frac{dh}{db}\). The derivative gives us the slope of the tangent line at any point \(b\) on the graph of the height function, meaning it provides the instantaneous rate at which the height increases as the base diameter grows.
The derivative of our function is:

• \(\frac{dh}{db} = 0.70 + 0.0134b\)

This tells us that:

• With each increase in \(b\), there is a minimum rate of increase in height of 0.70 meters per centimeter of base diameter.
• Additionally, the rate of change itself increases linearly with \(b\), as indicated by the 0.0134 coefficient of \(b\), meaning the growth in height becomes faster as the tree becomes wider.

Understanding this rate of change is crucial for predicting how factors like nutrient availability may further affect tree growth.

Graphing Functions

Graphing a function means plotting its values on a coordinate system to visualize the relationship between the variables. In this exercise, graphing enables us to see how the height of the conifer tree evolves as the base diameter changes.
The original function, \(h(b) = 5.67 + 0.70b + 0.0067b^2\), when graphed, will show a curve starting at \(5.67\) on the y-axis (tree height) when \(b = 0\) and rising as \(b\) increases. This visualization helps us grasp the quadratic nature, where the slope steepens as the base diameter grows.
Plotting the derivative \(\frac{dh}{db} = 0.70 + 0.0134b\) represents the rate of change on the graph. This plot is a straight line, indicating steady acceleration in height with wider base diameters, reflecting more rapid growth as trees mature.

• The slope of this line in the derivative plot represents how much faster the height is growing per unit increase in \(b\).

Through graphing, these abstract mathematical concepts become tangible, allowing learners to connect theoretical derivatives to observable patterns in growth.