Textbook Question
Find the unknown side lengths in each pair of similar triangles.
Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 2, Problem 23
Length of a Shadow If a tree 20 ft tall casts a shadow 8 ft long, how long would the shadow of a 30-ft tree be at the same time and place?
Verified step by step guidance1
Recognize that this problem involves similar triangles formed by the tree and its shadow, where the ratio of the height of the tree to the length of its shadow remains constant at the same time and place.
Set up the ratio using the given information: the height of the first tree (20 ft) over its shadow length (8 ft), which can be written as \(\frac{20}{8}\).
Let the length of the shadow of the 30-ft tree be \(x\). Set up the proportion \(\frac{20}{8} = \frac{30}{x}\) to express the equality of ratios.
Solve the proportion for \(x\) by cross-multiplying: \(20 \times x = 8 \times 30\).
Isolate \(x\) by dividing both sides by 20: \(x = \frac{8 \times 30}{20}\). This expression gives the length of the shadow of the 30-ft tree.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Similar Triangles
When two triangles have the same angles, they are similar, meaning their corresponding sides are proportional. In this problem, the tree and its shadow form a right triangle, and comparing two such triangles allows us to set up ratios to find unknown lengths.
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30-60-90 Triangles
Proportionality in Right Triangles
The lengths of sides in right triangles that share the same angle maintain a constant ratio. This principle lets us relate the height and shadow length of one tree to another under identical lighting conditions, enabling calculation of unknown shadow lengths.
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Application of Ratios
Ratios compare two quantities and are used to solve for unknown values in proportional relationships. Here, the ratio of the height to shadow length of the first tree helps determine the shadow length of the second tree by setting up and solving a proportion.
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Related Practice
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