Question

which of the following measurements are expressed to 5 sigfigs? 147250 cm; 43.270 × 10⁴ cm; 0.2005 cm; 3.45 × 10⁴ cm

Explanation:

Step1: Recall sig fig rules

Significant figures (sig figs) rules: Non - zero digits are significant. Zeros between non - zero digits are significant. Trailing zeros in a decimal number or in a number with a decimal point are significant. In scientific notation, only the coefficient counts for sig figs.

Step2: Analyze 147250 cm

147250: The trailing zero may or may not be significant. But here, without a decimal, the trailing zero is ambiguous. Wait, no, let's count: 1,4,7,2,5 are non - zero, the last zero: if it's a placeholder, it's not significant. Wait, maybe I made a mistake. Wait, 147250: let's see, the digits are 1,4,7,2,5,0. The zero at the end: if the number is written as 147250 without a decimal, the trailing zero is not significant. So number of sig figs: 5? Wait, 1,4,7,2,5: that's 5 non - zero digits, and the zero is a placeholder? Wait, no, maybe the problem considers 147250 as having 5 sig figs? Wait, no, let's check other options.

Step3: Analyze \(43.270\times10^{4}\) cm

In scientific notation, the coefficient is 43.270. The number of sig figs in the coefficient: 4,3,2,7,0. The zero is trailing after the decimal, so it's significant. So coefficient has 5 sig figs. So this number has 5 sig figs.

Step4: Analyze 0.2005 cm

0.2005: leading zero is not significant. Then 2,0,0,5. The zeros between 2 and 5 are significant. So number of sig figs: 4 (2,0,0,5)? Wait, no: 0.2005: digits are 2,0,0,5. Wait, 2 is first non - zero, then 0,0,5. So that's 4 sig figs? Wait, no, 0.2005: the number of sig figs is 4? Wait, no, 2,0,0,5: that's four digits. Wait, maybe I miscalculated.

Step5: Analyze \(3.45\times10^{4}\) cm

Coefficient is 3.45, which has 3 sig figs.

Wait, let's re - evaluate 147250: If we consider that the zero is a significant figure (maybe the number is precise to the unit, so 147250 has 5 sig figs: 1,4,7,2,5). And \(43.270\times10^{4}\): coefficient 43.270 has 5 sig figs (4,3,2,7,0). 0.2005: 2,0,0,5: 4 sig figs. \(3.45\times10^{4}\): 3 sig figs.

Wait, maybe the correct ones are 147250 cm (if we take the trailing zero as non - significant? No, that can't be. Wait, no, let's use the rules properly:

- For a number without a decimal: trailing zeros are not significant. For a number with a decimal: trailing zeros are significant.

- 147250: no decimal, so trailing zero is not significant. So sig figs: 5 (1,4,7,2,5).

- \(43.270\times10^{4}\): coefficient has a decimal, trailing zero is significant. So sig figs in coefficient: 5 (4,3,2,7,0).

- 0.2005: decimal present, leading zero not significant. Digits: 2,0,0,5. So 4 sig figs.

- \(3.45\times10^{4}\): coefficient has 3 sig figs.

So the measurements with 5 sig figs are 147250 cm and \(43.270\times10^{4}\) cm? Wait, but let's check 147250 again. If the number is 147250, and we assume that the zero is a significant figure (maybe the number is written as 147250. with a decimal, but it's not). Wait, maybe the problem has a different approach. Let's count the significant figures for each:

1. 147250 cm:
- Digits: 1, 4, 7, 2, 5, 0.
- The trailing zero: if it's a significant figure (e.g., the measurement is precise to the unit, so the zero is significant), then there are 6? No, that's not right. Wait, maybe the problem considers 147250 as having 5 sig figs (ignoring the trailing zero as a placeholder). So 1,4,7,2,5: 5 sig figs.

2. \(43.270\times10^{4}\) cm:
- Coefficient: 43.270.
- Sig figs in coefficient: 4, 3, 2, 7, 0 (the zero is significant because it's after the decimal). So 5 sig figs.

3. 0.2005 cm:
- Sig figs: 2, 0, 0, 5 (the leading zero is not significant, the two zeros between 2 and 5 are significant). So 4 sig figs.

4. \(3.45\times10^{4}\) cm:
- Coefficient: 3.45.
- Sig figs: 3, 4, 5. So 3 sig figs.

So the measurements with 5 significant figures are 147250 cm and \(43.270\times10^{4}\) cm. But let's check the options again. The green box: 147250 cm, blue box: \(43.270\times10^{4}\) cm, orange: 0.2005 cm, cyan: \(3.45\times10^{4}\) cm.

Wait, maybe I made a mistake with 147250. Let's use the rule: All non - zero digits are significant. Zeros between non - zero digits are significant. Trailing zeros in a number with a decimal are significant. Trailing zeros in a number without a decimal are ambiguous, but sometimes considered not significant. But in some cases, if the number is written as 147250, and we assume that the zero is a significant figure (maybe the measurement is to the nearest unit, so the zero is significant), then 147250 has 5 sig figs (1,4,7,2,5) and the zero is a placeholder? No, that's conflicting.

Alternatively, maybe the correct ones are 147250 cm (5 sig figs: 1,4,7,2,5) and \(43.270\times10^{4}\) cm (5 sig figs: 4,3,2,7,0). And 0.2005 has 4, \(3.45\times10^{4}\) has 3.

So the measurements expressed to 5 sig figs are 147250 cm and \(43.270\times10^{4}\) cm. But let's check the coefficient of \(43.270\times10^{4}\): 43.270 has 5 sig figs (the zero at the end is significant because it's after the decimal). So that's correct. And 147250: if we count 1,4,7,2,5 as significant (the zero is a placeholder), then it has 5 sig figs.

Answer:

147250 cm, \(43.270\times10^{4}\) cm