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Chapter 1: Problem 50

Refer to Problem \(48 .\) Suppose the mountain height is \(y\), the wonman's original distance from the mountain is \(x\), and the angle of elevation she measures from the horizontal to the top of the mountain is \(\theta .\) If she moves a distance \(d\) closer to the mountain and measures an angle of clevation \(\phi\), find a general equation for the height of the mountain y in terms of \(d, \phi\), and \(\theta\), neglecting the height of her eyes above the ground.

Short Answer

Expert verified
The height of the mountain \(y\) in terms of \(d\), \(\phi\), and \(\theta\) is given by \(y = \frac{d \cdot \tan(\phi)}{\tan(\phi) - \csc(\theta)}\).

Step by step solution

01

Deriving the equation for the first scenario

In the first scenario, the distance from the mountain is \(x\) and the angle measured is \(\theta\). Since this forms a right-angled triangle, we can use the tangent of the angle \(\theta\) for the triangle which is equal to the height \(y\) divided by the base \(x\). This gives us the equation \(y = x \cdot \tan(\theta)\)
02

Deriving the equation for the second scenario

In the second scenario, the woman has moved closer to the mountain. The distance from the mountain is now \(x - d\), where \(d\) is the distance moved. She measures a new angle of \(\phi\). Using these values in the definition of tangent for the new triangle, we obtain \(y = (x - d) \cdot \tan(\phi)\)
03

Manipulating the equations to express \(y\) in terms of \(d\), \(\phi\), and \(\theta\)

We have two equations \(y = x \cdot \tan(\theta)\) and \(y = (x - d) \cdot \tan(\phi)\) to solve for \(y\). Substituting \(x\) from the first equation into the second gives us \(y = (y \cdot \csc(\theta) - d) \cdot \tan(\phi)\). Separating terms then gives \(y \tan(\phi) = y \cdot \csc(\theta) \cdot \tan(\phi) - d \cdot \tan(\phi)\), which simplifies to \(y(1 - \csc(\theta) \cdot \tan(\phi)) = -d \cdot \tan(\phi)\). Solving the resulting equation for \(y\) will give the required general equation for the mountain height.
04

Final equation for the mountain height

The final equation for the mountain height is consequently \(y = \frac{d \cdot \tan(\phi)}{\tan(\phi) - \csc(\theta)}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle of Elevation
The angle of elevation is a key concept in trigonometry, especially when it comes to applying it in physics and real-world contexts like estimating the height of a mountain, a building, or any other tall object. It refers to the angle between the horizontal line of sight and the line of sight directed upwards towards an object. When you look up at the peak of a mountain, for instance, the angle your line of sight makes with the horizontal ground is the angle of elevation.

Understanding the angle of elevation is critical as it's intimately connected to right-angled triangles formed between the observer, the top of the object being observed, and the point directly beneath the object's peak on the ground level. When the angle of elevation and the horizontal distance to the object are known, one can calculate the object's height using trigonometric functions.
Right-Angled Triangle
A right-angled triangle is a triangle in which one of the angles is exactly 90 degrees. This type of triangle is the foundation for many trigonometric applications. The sides of a right-angled triangle have specific names: the side opposite the right angle is the 'hypotenuse,' the side opposite the angle of interest is the 'opposite,' and the side next to the angle of interest is the 'adjacent.'

In our context, the height of the mountain (\( y \)) forms the opposite side, while the horizontal distance from the mountain (\( x \) or \( x - d \)) is the adjacent side. The hypotenuse would be the line of sight from the observer to the top of the mountain. These elements are key in setting up equations using trigonometric functions to solve for unknown values such as the height of the mountain.
Tangent Function
The tangent function is one of the six fundamental trigonometric functions and is particularly useful when dealing with right-angled triangles. For a given angle \( \theta \), the tangent of \( \theta \) (\( \tan(\theta) \)) is defined as the ratio of the length of the opposite side to the length of the adjacent side of the right-angled triangle. This makes it perfect for calculating heights and distances.

In the exercise, you can see the tangent function in action as it is used to derive an expression for the mountain's height from the known distances and angles of elevation. The function essentially provides a direct relationship between the angle of elevation and the ratio of the mountain's height to the distance from it. This is crucial for students to grasp, as it translates the abstract concept of an angle into physical dimensions, enabling them to solve real-world problems.

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Most popular questions from this chapter

Two points in a rectangular coordinate system have the coordinates \((5.0,3.0)\) and \((-3.0,4.0)\), where the units are centimeters. Determine the distance between these points:

A woman measures the angle of elevation of a mountaintop as \(12.0^{\circ}\). After walking \(1.00 \mathrm{~km}\) closer to the mountain on level ground, she finds the angle to be \(14.0^{\circ} .\) (a) Draw a picture of the problem, neglecting the height of the woman's eyes above the ground. Hint: Use two triangles. (b) Select variable names for the mountain height (suggestion: \(y\) ) and the woman's original distance from the mountain (suggestion: \(x\) ) and label the picture. (c) Using the labeled picture and the tangent function, write two trigonometric equations relating the two sclected variables. (d) Find the height \(y\) of the mountain by first solving one equation for \(x\) and substituting the result into the other equation.

(a) About how many microorganisms are found in the human intestinal tract? (A typical bacterial length scale is \(10^{-6} \mathrm{~m}\). Estimate the intestinal volume and assume one hundredth of it is occupied by bacteria.) (b) Discuss your answer to part (a). Are these bacteria beneficial, dangerous, or neutral? What finctions could they serve?

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A fisherman catches two striped bass. The smaller of the two has a measured length of \(93.46 \mathrm{~cm}\) (two decimal places, four significant figures), and the larger fish has a measured length of \(135.3 \mathrm{~cm}\) (one decimal place, four significant figures). What is the total length of fish caught for the day?
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