- Identify the first two significant figures and use the third to round.
- Convert each result into standard form \(a \times 10^n\).
- The results are:
A\(\boxed{8.9 \times 10^{11}}\)
B\(\boxed{1.0 \times 10^6}\)
C\(\boxed{2.9 \times 10^7}\)
D\(\boxed{1.3 \times 10^{-3}}\)
E\(\boxed{3.0 \times 10^7}\)
f. \(\boxed{1.3 \times 10^{-3}}\)
Explanation
- Understanding Significant Figures and Standard Form
To round a number to two significant figures, we identify the first two non-zero digits. The third digit determines whether to keep the second digit as is (if the third digit is less than 5) or to round it up (if the third digit is 5 or greater). After rounding, we express the result in standard form, which is \(a \times 10^n\), where \(1 \le |a| < 10\) and \(n\) is an integer.
- Performing the Calculations for Each Item
AThe budget is \(\#892.46\) billion.
892.46 billion is \(892,460,000,000\). The first two significant figures are 8 and 9. The third digit is 2, which is less than 5, so we keep 8.9. In standard form, this is \(\boxed{8.9 \times 10^{11}}\).
BThe number of orphans is \(995,000\).
The first two significant figures are 9 and 9. The third digit is 5, so we round the second 9 up, which changes 99 to 100. Thus, we have \(1.0 \times 10^2 \times 10^4 = 1.0 \times 10^6\). In standard form, this is \(\boxed{1.0 \times 10^6}\).
CThe value of imports is \(\#28,500,000\).
The first two significant figures are 2 and 8. The third digit is 5, so we round up the 8 to 9, giving 2.9. In standard form, this is \(\boxed{2.9 \times 10^7}\).
DThe density of air is \(0.001293 \text{ g.cm}^{-3}\).
The first two significant figures are 1 and 2. The third digit is 9, so we round up the 2 to 3, giving 1.3. The decimal point moves 3 places to the right. In standard form, this is \(\boxed{1.3 \times 10^{-3}}\).
EThe land area is \(30,280,000 \text{ km}^2\).
The first two significant figures are 3 and 0. The third digit is 2, which is less than 5, so we keep 3.0. In standard form, this is \(\boxed{3.0 \times 10^7}\).
f. 1 watt is \(0.001341\) horsepower.
The first two significant figures are 1 and 3. The third digit is 4, which is less than 5, so we keep 1.3. In standard form, this is \(\boxed{1.3 \times 10^{-3}}\).
Examples
Rounding to significant figures and using standard form is essential in fields like science, engineering, and economics where handling very large or very small numbers is common. For instance, an astronomer measuring the distance between galaxies (which can be millions of light-years) or a chemist measuring the mass of a microscopic molecule needs a consistent way to record measurements. By using two significant figures, they maintain the level of precision appropriate for their tools, and by using standard form, they can easily compare the magnitudes of different measurements without worrying about counting trailing or leading zeros.