Problem 86 - Algebra and Trigonometry Real Mathematics, Real People

Question

Meteorology A meteorologist is positioned 100 feet from the point at which a weather balloon is launched. When the balloon is at height \(h,\) the distance \(d\) (in feet) between the meteorologist and the balloon is given by \(d=\sqrt{100^{2}+h^{2}}\) (a) Use a graphing utility to graph the equation. Use the trace feature to approximate the value of \(h\) when \(d=200.\) (b) Complete the table. Use the table to approximate the value of \(h\) when \(d=200.\) $$\begin{array}{|c|c|c|c|c|c|c|} \hline h & 160 & 165 & 170 & 175 & 180 & 185 \\ \hline d & & & & & & \\ \hline \end{array}$$ (c) Find \(h\) algebraically when \(d=200.\) (d) Compare the results of each method. In each case, what information did you gain that wasn't revealed by another solution method?

Step-by-Step Solution

Answer

The height of the balloon when the distance \(d=200\) is approximately \(h = 173.21\) feet.

Step 1: (a) Graph the equation

Firstly, you should graph the equation \(d= \sqrt{100^{2}+h^{2}}\) using a graphing utility. Then, trace the curve to find the height \(h\) when \(d=200\). By using a graphing utility to perform this task, at \(d=200\), the corresponding value of \(h\) will be around 100.

Step 2: (b) Complete the table

You need to substitute each value of \(h\) given in the table into the equation then solve for \(d\). The resulting table will look somewhat like this: \[\begin{array}{|c|c|c|c|c|c|c|} \hline h & 160 & 165 & 170 & 175 & 180 & 185 \ \hline d & 195.41 & 203.91 & 212.46 & 221.07 & 229.71 & 238.41 \ \hline \end{array}\]By the above table, we can see that the value of \(h\) will be roughly between 165 and 170 when \(d=200\).

Step 3: (c) Find the value of \(h\) algebraically

To find \(h\) algebraically when \(d=200\), we just substitute \(d=200\) into the given equation and solve for \(h\), as follow:Squaring both sides, we get: \[200^2 = 100^2 + h^2 \implies 30000 = h^2 \implies h = \sqrt{30000} \approx 173.21\]

Step 4: (d) Compare the results of each method

By comparing the results from the graphical, tabular, and algebraic solutions, we can conclude that they are around the same for \(h\) when \(d=200\). The graphical and tabular methods give rough estimates while the algebraic method provides a more accurate result. Each method can give different insights to the problem. It can be seen the role of height on the distance in each method, graphically, algebraically, and in a table

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry that relates the lengths of the sides of a right triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is represented as: \

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• \(a^2 + b^2 = c^2\)
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\In the context of the meteorology exercise, this theorem is used to calculate the distance \(d\) from the meteorologist to the weather balloon, with \(100\) feet being the horizontal distance and \(h\) the height of the balloon. By applying the Pythagorean Theorem: \

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• \(d = \sqrt{100^2 + h^2}\)
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\Understanding this equation helps in visualizing how the concept of right triangles is applied in real-world problems, where the distance between two points is calculated geometrically.

Graphing Utility

A graphing utility is a tool, often computer software or a physical calculator, that allows users to explore the graphs of mathematical equations visually. It is particularly helpful in solving problems where intuitive understanding through visual representation is beneficial. \In the given exercise, students are prompted to use a graphing utility to graph the equation \(d = \sqrt{100^2 + h^2}\). This graphical representation of the equation allows students to trace and approximate various distances \(d\) corresponding to different heights \(h\). By observing the curve, they can see the relationship between the balloon's height and the distance traveled. For instance, the graphing utility can be used to find that \(h\) is approximately \(100\) when \(d = 200\). \Using a graph encourages intuitive learning and helps students relate abstract mathematical formulas to tangible results.

Algebraic Solution

An algebraic solution involves solving equations using mathematical operations and properties. In this exercise, solving for \(h\) when \(d = 200\) can be done by reconnecting the problem directly through algebra. \To solve algebraically, you set the equation: \

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• \(200^2 = 100^2 + h^2\)
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• \(40000 = 10000 + h^2\)
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• \(30000 = h^2\)
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• \(h = \sqrt{30000} \approx 173.21\)
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\This approach provides precise values for \(h\), unlike estimations from graphical methods. Algebraic solutions are essential for accuracy, offering insights that graphical and tabular methods may only approximate. The confidence in precision is what sets algebra apart in mathematical analysis.

Mathematical Modeling

Mathematical modeling involves using mathematics to represent, analyze, make predictions, or otherwise provide insight into real-world phenomena. In the meteorology problem, the given equation \(d = \sqrt{100^2 + h^2}\) is a mathematical model that depicts the relationship between the meteorologist and the weather balloon based on geometric principles. \Modeling allows complex real-world situations to be simplified into manageable mathematical expressions that can be solved and interpreted more easily. \

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• Helps predict outcomes like the balloon's height at a certain distance \(d\).
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• Enables comparison of various solution methods as seen in the exercise.
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\Through mathematical modeling, students learn to see the interconnectedness of mathematics with the real world, aiding their ability to apply mathematical reasoning to diverse scenarios.