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Chapter 1: Problem 66

While standing in the Mall in Washington, D.C., a tourist observes the angle of elevation to the top of the Washington Monument to be \(67^{\circ}\). After moving \(1012 \mathrm{ft}\) farther away from the Washington Monument, the angle of elevation changes to \(24^{\circ}\). a) Use the small triangle to find \(x\) in terms of \(h\). b) Use the large triangle to find the height of the Washington Monument.

Short Answer

Expert verified
The height of the Washington Monument is approximately equal to \( h = \frac{1012 \tan(24^{\backslashcirc}) \tan(67^{\backslashcirc})}{\tan(67^{\backlashcirc}) - \tan(24^{\backlashcirc})} \).

Step by step solution

01

- Understand the Problem

We need to find the height of the Washington Monument using the given angles of elevation and distances.
02

- Identify the Right Triangles

There are two right triangles: a small triangle formed at the initial position and a large triangle formed after moving 1012 ft away.
03

- Use the Tangent Function for Each Triangle

For the small triangle: \( \tan(67^{\backslashcirc}) = \frac{h}{x} \) For the large triangle: \( \tan(24^{\backslashcirc}) = \frac{h}{x + 1012} \)
04

- Solve for x in Terms of h (Small Triangle)

Rearrange the equation \( \tan(67^{\backslashcirc}) = \frac{h}{x} \) to find \( x = \frac{h}{\tan(67^{\backslashcirc})} \)
05

- Substitute x into the Equation from Large Triangle

Use the large triangle equation \( \tan(24^{\backslashcirc}) = \frac{h}{x + 1012} \). Substitute \( x \) with \( \frac{h}{\tan(67^{\backslashcirc})} \), giving \( \tan(24^{\backslashcirc}) = \frac{h}{\frac{h}{\tan(67^{\backslashcirc})} + 1012} \)
06

- Solve for h

Simplify and solve the resulting equation: \[ \tan(24^{\backslashcirc}) = \frac{h \tan(67^{\backslashcirc})}{h + 1012 \tan(67^{\backslashcirc})} \] Cross-multiply to get: \[ h \tan(67^{\backslashcirc}) = \tan(24^{\backslashcirc}) (h + 1012 \tan(67^{\backslashcirc})) \] Distribute and collect terms: \[ h \tan(67^{\backslashcirc}) = h \tan(24^{\backslashcirc}) + 1012 \tan(24^{\backslashcirc}) \tan(67^{\backslashcirc}) \] Isolate \( h \): \[ h (\tan(67^{\backslashcirc}) - \tan(24^{\backslashcirc})) = 1012 \tan(24^{\backslashcirc}) \tan(67^{\backslashcirc}) \] \[ h = \frac{1012 \tan(24^{\backslashcirc}) \tan(67^{\backslashcirc})}{\tan(67^{\backslashcirc}) - \tan(24^{\backslashcirc})} \]
07

- Calculate the Height

Use a calculator to evaluate \( \tan(67^{\backslashcirc}) \) and \( \tan(24^{\backslashcirc}) \), then compute \( h \).

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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 the angle formed between the horizontal ground and the line of sight to an object above the ground. In our problem, the tourist observes the Washington Monument at two different angles: first at 67掳 and then at 24掳 after moving farther away. Understanding and calculating the angle of elevation helps us determine the height of tall structures.
right triangle
A right triangle has one angle exactly equal to 90掳, which simplifies calculations in trigonometry. In our problem, the tourist鈥檚 points of observation create two right triangles with the Washington Monument. These triangles help us form relationships between the height of the monument (h) and the horizontal distances from the observation points (x and x + 1012 ft). This breakdown into right triangles helps simplify our calculations.
tangent function
The tangent function (tan) is a crucial trigonometric function for solving problems involving right triangles. It鈥檚 defined as the ratio of the opposite side to the adjacent side in a right triangle. For the smaller right triangle, we have: \[ \tan(67^{\circ}) = \frac{h}{x} \] And for the larger right triangle: \[ \tan(24^{\circ}) = \frac{h}{x + 1012} \] The tangent function helps us establish equations to relate the height of the monument with the distances x and x + 1012 ft, ultimately aiding in our goal to find the height of the monument (h).
solving equations
Solving the equations derived from the tangent functions involves algebraic manipulation. First, we solve for x in terms of h from the small triangle: \[ x = \frac{h}{\tan(67^{\circ})} \] Next, we substitute this x value into the equation for the large triangle and solve for h: \[ \tan(24^{\circ}) = \frac{h}{\frac{h}{\tan(67^{\circ})} + 1012} \] This leads to: \[ h(\tan(67^{\circ}) - \tan(24^{\circ})) = 1012 \tan(24^{\circ}) \tan(67^{\circ}) \ h = \frac{1012 \tan(24^{\circ}) \tan(67^{\circ})}{\tan(67^{\circ}) - \tan(24^{\circ})} \] Solving these equations step by step helps find the height h of the Washington Monument, demonstrating the application of algebra in trigonometric contexts.

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Most popular questions from this chapter

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