Textbook Question
Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) I , r/y
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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 50
Solve each problem. Height of a Lunar Peak The lunar mountain peak Huygens has a height of 21,000 ft. The shadow of Huygens on a photograph was 2.8 mm, while the nearby mountain Bradley had a shadow of 1.8 mm on the same photograph. Calculate the height of Bradley. (Data from Webb, T., Celestial Objects for Common Telescopes, Dover Publications.)
Verified step by step guidance1
Understand that the photograph creates a scale model where the actual heights of the mountains are proportional to the lengths of their shadows on the photograph.
Set up a proportion relating the height of Huygens to its shadow length and the height of Bradley to its shadow length: \(\frac{\text{Height of Huygens}}{\text{Shadow of Huygens}} = \frac{\text{Height of Bradley}}{\text{Shadow of Bradley}}\).
Substitute the known values into the proportion: \(\frac{21000}{2.8} = \frac{\text{Height of Bradley}}{1.8}\).
Solve the proportion for the height of Bradley by cross-multiplying: \(\text{Height of Bradley} = \frac{21000 \times 1.8}{2.8}\).
Calculate the value from the expression above to find the height of Bradley.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Similar Triangles and Proportionality
When two objects cast shadows under the same lighting conditions, their heights and shadow lengths form similar triangles. This means the ratio of an object's height to its shadow length is constant, allowing us to find unknown heights by setting up proportions.
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30-60-90 Triangles
Ratio and Proportion
Ratio compares two quantities, and proportion states that two ratios are equal. In this problem, the ratio of height to shadow length for Huygens and Bradley can be set equal, enabling calculation of Bradley's height using the known values.
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6:04Introduction to Trigonometric Functions
Unit Consistency and Conversion
Ensuring consistent units is crucial when solving problems involving measurements. Here, the shadows are measured in millimeters while heights are in feet, so the ratio uses the same units for shadow lengths to maintain accuracy in calculations.
Recommended video:
06:11Introduction to the Unit Circle
Related Practice
Textbook Question
Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) I , y/r
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Textbook Question
Perform each calculation. See Example 3. 90° ― 36° 18' 47"
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Textbook Question
Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3.
csc θ > 0 , cot θ > 0
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Textbook Question
Perform each calculation. See Example 3. 90° ― 17° 13'
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Textbook Question
Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3.
tan θ < 0 , cot θ < 0
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