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Chapter 1: Problem 67

According to the rules on significant figures, the product of the measured quantities \(99.9 \mathrm{m}\) and \(1.008 \mathrm{m}\) should be expressed to three significant figures-\(101 \mathrm{m}^{2} .\) Yet, in this case, it would be more appropriate to express the result to four significant figures-\(100.7 \mathrm{m}^{2} .\) Explain why.

Short Answer

Expert verified
In this case, it would be more appropriate to express the result to four significant figures because the rules of significant figures challenge the precision of our initial measurements when three significant figures are used. So, to represent the accuracy of the used values, it's better to keep an extra significant figure in the final answer.

Step by step solution

01

Understand the rules of significant figures in calculations

When multiplying or dividing, the result should have the same number of significant figures as the number in the calculation with the fewest significant figures. In our case, \(99.9 \mathrm{m}\) has three significant figures and \(1.008 \mathrm{m}\) has four. So, according to this rule, the result should be expressed with three significant figures.
02

Calculate the Product

When we multiply these two numbers, we get \(99.9 \mathrm{m} * 1.008 \mathrm{m} = 100.792 \mathrm{m}^{2}\). The number 100.792 has six significant figures.
03

Apply the rule of significant figures

According to the rules of significant figures in multiplication, our answer should be rounded to three significant figures, which gives us \(101 \mathrm{m}^{2}\).
04

Analyze the discrepancy

In this case, however, providing the answer with three significant figures is misleading because it suggests a degree of inaccuracy in the initial measurements that was not there. The decimal places in the numbers 99.9m and 1.008m imply that these measurements were quite precise, and so a three-significant-figure answer implies a level of uncertainty that isn’t justified by the data.
05

Conclude

Therefore, in situations where the significant figures rules might suggest reporting a measurement that falsely implies a high level of uncertainty, it might be more appropriate to keep an extra significant figure in the final answer.

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Most popular questions from this chapter

The non-SI unit, the hand (used by equestrians), is 4 inches. What is the height, in meters, of a horse that stands 15 hands high?

The reported value for the volume of a rectangular piece of cardboard with the dimensions \(36 \mathrm{cm} \times\) \(20.2 \mathrm{cm} \times 9 \mathrm{mm}\) should be \((\mathrm{a}) \quad 6.5 \times 10^{3} \mathrm{cm}^{3};\) (b) \(7 \times 10^{2} \mathrm{cm}^{3} ;\) (c) \(655 \mathrm{cm}^{3} ;\) (d) \(6.5 \times 10^{2} \mathrm{cm}^{3}\).

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