Question

question 6 (5 points)
determine the correct number of significant figures in each of the following: use the list to answer. they may be used more than once.

column a

1. ____ 3427
2. ____ 10,000,000
3. ____ 3100.0 x 10²
4. ____ 0.0000455
5. ____ 0.902 x 10⁻³
6. ____ 0.005650
7. ____ 120,860

column b
a. 1 significant figure
b. 9 significant figures
c. 4 significant figures
d. 3 significant figures
e. 6 significant figures
f. 7 significant figures
g. 5 significant figures
h. 8 significant figures

Answer:

To solve for the number of significant figures in each value, we use the rules of significant figures:

• Non - zero digits are always significant.
• Zeros between non - zero digits are significant.
• Trailing zeros in a number with a decimal point are significant.
• Leading zeros (zeros before the first non - zero digit) are not significant.
• For numbers in scientific notation, we only consider the significant figures in the coefficient.

1. For 3427

All the digits (3, 4, 2, 7) are non - zero. So the number of significant figures is 4.

2. For 10,000,000

The trailing zeros here are not significant (since there is no decimal point). Only the digit 1 is significant. So the number of significant figures is 1.

3. For \(3100.0\times10^{2}\)

First, consider the coefficient \(3100.0\). The non - zero digits 3 and 1, the zero between them, and the trailing zero after the decimal are significant. So the number of significant figures in the coefficient is 5? Wait, no. Wait, \(3100.0\): the digits are 3, 1, 0, 0, 0? No, wait \(3100.0\) has digits 3, 1, 0, 0, 0? No, \(3100.0\) is 3 (thousands place), 1 (hundreds place), 0 (tens place), 0 (ones place), and 0 (tenths place). Wait, no, the correct way: non - zero digits (3,1) are significant, the zeros between non - zeros? No, here after 3 and 1, we have two zeros and then a zero after the decimal. Wait, the rule for trailing zeros after a decimal: in \(3100.0\), the zeros after the decimal (the last 0) is significant, and the zeros between 1 and the decimal: wait, \(3100.0\) can be thought of as \(3.1000\times10^{3}\)? No, the original number is \(3100.0\times10^{2}=310000\). But when we look at \(3100.0\), the digits are 3,1,0,0,0? No, \(3100.0\) has five significant figures? Wait, no. Wait, the number \(3100.0\): the 3,1 are non - zero, the two zeros between 1 and the decimal, and the zero after the decimal. Wait, the correct count: non - zero digits (3,1) are significant, the zero between 3 and 1? No, 3 and 1 are separated by two zeros? Wait, no, \(3100.0\) is 3 (thousand), 1 (hundred), 0 (ten), 0 (one), 0 (tenth). So the significant figures are 3,1,0,0,0? No, that's not right. Wait, the rule is: trailing zeros in a number with a decimal are significant. So in \(3100.0\), the zeros after the decimal (the last 0) is significant, and the zeros between 1 and the decimal: are they significant? Wait, if there is a decimal, then all the digits from the first non - zero digit are significant. So \(3100.0\): the first non - zero digit is 3, then 1, then 0, 0, 0? No, \(3100.0\) has five significant figures? Wait, no, let's re - calculate. \(3100.0\): the digits are 3,1,0,0,0? No, \(3100.0\) is 3 (1000's place), 1 (100's place), 0 (10's place), 0 (1's place), 0 (0.1's place). So the significant figures are 3,1,0,0,0? No, that's 5? Wait, no, the correct answer is that \(3100.0\) has 5 significant figures? Wait, no, let's check the rules again. The rules for significant figures:

• All non - zero digits are significant.
• Zeros between non - zero digits are significant.
• Trailing zeros in a number with a decimal point are significant.
• Leading zeros are not significant.

So for \(3100.0\):

• Non - zero digits: 3,1 (2 digits)
• Zeros between non - zero digits: none
• Trailing zeros after decimal: 1 zero (the last 0)
• Zeros between 1 and the decimal: two zeros. Wait, are these zeros significant? Since there is a decimal at the end, all the digits from the first non - zero digit are significant. So \(3100.0\) has 5 significant figures? Wait, no, \(3100.0\) is equal to \(3.1000\times10^{3}\), so the coefficient \(3.1000\) has 5 significant figures. So the number of significant figures is 5? Wait, but in the options, option g is 5 significant figures. Wait, maybe I made a mistake earlier. Let's re - do:

Wait, \(3100.0\times10^{2}\) is in the form of a number with a decimal and a power of 10. The significant figures are determined by the number before the power of 10. \(3100.0\) has 5 significant figures (3,1,0,0,0? No, 3,1,0,0,0? Wait, 3 (1), 1 (2), 0 (3), 0 (4), 0 (5)? No, that's not correct. Wait, \(3100.0\): the digits are 3,1,0,0,0? No, \(3100.0\) is 3 (thousand), 1 (hundred), 0 (ten), 0 (one), 0 (tenth). So the significant figures are 3,1,0,0,0? No, the zero after the decimal is significant, and the zeros between 1 and the decimal: since there is a decimal, those zeros are significant. So \(3100.0\) has 5 significant figures. So the answer for 3 is g.

4. For 0.0000455

Leading zeros are not significant. The non - zero digits are 4,5,5. So the number of significant figures is 3.

5. For \(0.902\times10^{- 3}\)

In scientific notation, we look at the coefficient \(0.902\). The non - zero digit 9, the zero between 9 and 2, and the digit 2 are significant. So the number of significant figures is 3.

6. For 0.005650

Leading zeros are not significant. The non - zero digits 5,6,5 and the trailing zero after 5 are significant. So the number of significant figures is 4? Wait, no: 0.005650. The leading zeros (three zeros) are not significant. Then we have 5,6,5,0. The zero at the end is significant (because it's after the decimal). So the significant figures are 5,6,5,0. So that's 4 significant figures? Wait, no, 5 (1), 6 (2), 5 (3), 0 (4). So 4 significant figures. So the answer is c.

7. For 120.860

The digits are 1,2,0,8,6,0. The non - zero digits 1,2,8,6 are significant. The zero between 2 and 8 is significant (because it's between non - zero digits), and the trailing zero after 6 is significant (because there is a decimal). So the number of significant figures is 6? Wait, 1 (1), 2 (2), 0 (3), 8 (4), 6 (5), 0 (6). So 6 significant figures. So the answer is e.

Final Answers:

1. 3427: c. 4 significant figures
2. 10,000,000: a. 1 significant figure
3. \(3100.0\times10^{2}\): g. 5 significant figures
4. 0.0000455: d. 3 significant figures
5. \(0.902\times10^{-3}\): d. 3 significant figures
6. 0.005650: c. 4 significant figures
7. 120.860: e. 6 significant figures