Textbook Question
In Exercises 29–36, find the length x to the nearest whole unit.
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2. Trigonometric Functions on Right Triangles
Solving Right Triangles
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Textbook Question
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.)
Verified step by step guidance1
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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Video duration:
2mKey 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.
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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.
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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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