Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Forest growth Temperature and precipitation have a significant effect on plant life. If either the average annual temperature or the amount of precipitation is too low, trees and forests cannot grow. Instead, only grasslands and deserts will exist. The relationship between average annual temperature \(T\) (in \({ }^{\circ} \mathrm{F}\) ) and average annual precipitation \(P\) (in inches) is a linear inequality. In order for forests to grow in a region, \(T\) and \(P\) must satisfy the inequality \(29 T-39 P<450\), where \(33 \leq T \leq 80\) and \(13 \leq P \leq 45\). (a) Determine whether forests can grow in Winnipeg, where \(T=37^{\circ} \mathrm{F}\) and \(P=21.2 \mathrm{in} .\) (b) Graph the inequality, with \(T\) on the horizontal axis and \(P\) on the vertical axis. (c) Identify the region on the graph that represents where forests can grow.

Step-by-Step Solution

Answer

(a) Forests can grow in Winnipeg. (b, c) Graph the inequality and shade below the line to show the forest region.

Step 1: Check Forest Growth Condition in Winnipeg

The inequality for forest growth is given by \(29T - 39P < 450\). To determine if forests can grow in Winnipeg, substitute \(T = 37\) and \(P = 21.2\) into the inequality:\[29(37) - 39(21.2) < 450\]Calculate each term:- \(29 \times 37 = 1073\)- \(39 \times 21.2 = 826.8\)Substitute these values into the inequality:\[1073 - 826.8 < 450\]Simplify:\[246.2 < 450\]Since this condition is satisfied, forests can grow in Winnipeg.

Step 2: Graph the Inequality

Graph the inequality \(29T - 39P < 450\), where \(33 \leq T \leq 80\) and \(13 \leq P \leq 45\).1. Rewrite the inequality in slope-intercept form by isolating \(P\): - Start with \(29T - 39P = 450\). - Rearrange to get \(-39P = 450 - 29T\). - Divide each term by \(-39\): \[P = \frac{29}{39}T - \frac{450}{39}\]2. The line \(P = \frac{29}{39}T - \frac{450}{39}\) forms the boundary. Since the inequality is \(<\), this is a dashed line on the graph.3. Plot this line on a coordinate plane with \(T\) on the x-axis (horizontal) and \(P\) on the y-axis (vertical).

Step 3: Identify Region of Forest Growth

1. The inequality \(29T - 39P < 450\) indicates the region below the boundary line (from Step 2) is where forests can grow.2. Shade the area below the dashed line to denote this region.3. Ensure the shaded region falls within the bounds of \(33 \leq T \leq 80\) and \(13 \leq P \leq 45\). This shaded region represents where forests can potentially grow.

Key Concepts

Forest growth conditions

The conditions necessary for forests to grow are heavily influenced by temperature and precipitation. For a region to support forest life, both of these elements must reach levels sufficient to sustain tree growth. If they are too low, the area may only support grasslands or deserts. This relationship can be expressed as a linear inequality, which in this context is given as \(29T - 39P < 450\), where \(T\) represents the temperature in degrees Fahrenheit and \(P\) the precipitation in inches.
By examining this inequality, we can determine whether specific locations can support forests. For example, in Winnipeg, with a temperature of \(37^{\circ} \mathrm{F}\) and precipitation of \(21.2\) inches, the inequality holds true, indicating optimal conditions for forest growth.
Understanding these conditions is essential for predicting vegetation types and planning environmental conservation efforts.

Graphing inequalities

To fully understand the relationship between temperature, precipitation, and forest growth, it's crucial to visualize the inequality \(29T - 39P < 450\) on a graph. Graphing inequalities helps in identifying the regions where specific conditions are met or not met.
To graph linear inequalities, you need to follow a systematic approach:

• First, convert the inequality into the slope-intercept form, if needed.


• Next, plot the line which forms the boundary of the inequality. For the inequality to hold true, the region of interest (where forests can grow) is below this line.


• If the inequality is \(<\) or \(>\), the boundary is a dashed line, signifying the boundary is not included in the solution set.


• Shade the area of the graph that satisfies the inequality, ensuring it lies between the specified bounds \(33 \leq T \leq 80\) and \(13 \leq P \leq 45\).

This shaded area visually presents the conditions where forests have the potential to thrive.

Slope-intercept form

The slope-intercept form of a line is a common way to express linear equations, and it makes graphing straightforward. This form is written as \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept.
In the context of our inequality, we rearranged the equation \(29T - 39P = 450\) into a slope-intercept form: \[P = \frac{29}{39}T - \frac{450}{39}.\]
This transformation involves rearranging terms to solve for \(P\), which is the variable on the vertical axis. The slope \(\frac{29}{39}\) indicates the degree of change in precipitation needed for each unit of change in temperature.
The y-intercept \(-\frac{450}{39}\) represents the precipitation level when the temperature is zero, though it's more of a theoretical point here since the temperature won’t realistically reach zero in this context. Understanding this form allows us to efficiently graph the inequality and link mathematical concepts to real-world scenarios.

Coordinate plane

The coordinate plane is a powerful tool for representing linear relationships graphically. It consists of two perpendicular axes: the horizontal axis (often the x-axis) and the vertical axis (often the y-axis). In our scenario, temperature \(T\) is plotted on the horizontal axis, while precipitation \(P\) is plotted on the vertical axis.
Using this two-dimensional space, we can visually explore the relationship between the two variables by plotting the linear inequality.

• Begin by marking the boundary line that results from the slope-intercept equation.


• Identify and shade the region that satisfies our inequality \(29T - 39P < 450\).


• Pay close attention to the constraints \(33 \leq T \leq 80\) and \(13 \leq P \leq 45\), ensuring the shaded area representing forest growth is restricted to these bounds.

By using the coordinate plane, we gain a visual appreciation of data and conditions, making it easier to understand abstract mathematical concepts and their practical applications.