Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 8

Chapter 3, Problem 8

Concept Check Refer to the discussion of accuracy and significant digits in this section to answer the following. Mt. Everest When Mt. Everest was first surveyed, the surveyors obtained a height of 29,000 ft to the nearest foot. State the range represented by this number. (The surveyors thought no one would believe a measurement of 29,000 ft, so they reported it as 29,002.) (Data from Dunham, W., The Mathematical Universe, John Wiley and Sons.)

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Understand that the height 29,000 ft is given to the nearest foot, which means the actual height could be slightly less or more than 29,000 ft but within a certain range.

Since the measurement is rounded to the nearest foot, the smallest possible value is 29,000 ft minus half a foot, and the largest possible value is 29,000 ft plus half a foot.

Express this range mathematically as: \(29,000 - 0.5 \leq \text{height} < 29,000 + 0.5\).

Simplify the inequality to get the range: \(28,999.5 \leq \text{height} < 29,000.5\) feet.

Note that the surveyors reported 29,002 ft instead of 29,000 ft to make the measurement seem more believable, but the actual range based on rounding remains as calculated.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Significant Digits and Measurement Precision

Significant digits indicate the precision of a measurement, showing which digits are known reliably. A measurement reported as 29,000 ft to the nearest foot implies the last digits are uncertain, affecting the range of possible true values. Understanding significant digits helps interpret the accuracy and reliability of reported data.

Range of Values Represented by a Measurement

The range of a measurement defines the interval within which the true value lies, based on the precision of the measurement. For example, 29,000 ft to the nearest foot means the actual height could be from 28,999.5 ft up to 29,000.5 ft. This concept is essential to express uncertainty in measurements.

Rounding and Reporting Measurements

Rounding involves adjusting a number to a specified precision, which can affect how measurements are reported and interpreted. Surveyors may round values for simplicity or credibility, as in reporting 29,002 ft instead of 29,000 ft, which can influence perceived accuracy and the communicated range of the measurement.

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Concept Check Match each angle in Column I with its reference angle in Column II. Choices may be used once, more than once, or not at all. See Example 1. I. II. 5. A. 45° 6. B. 60° 7. -135° C. 82° 8. D. 30° 9. E. 38° 10. F. 32°

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Concept Check Match each angle in Column I with its reference angle in Column II. Choices may be used once, more than once, or not at all. See Example 1. I. II. 5. A. 45° 6. B. 60° 7. C. 82° 8. D. 30° 9. E. 38° 10. 480° F. 32°

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Determine whether each statement is true or false. If false, tell why. tan 60° ≥ cot 40°

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CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.

Column I: 1.

scs 80°

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°