Problem set 1 significant figures Answer Sheet - Dr. Behrang ... - Free Printable

Since the user has not selected any of the options (ExplainAnswer, TeachTopic, GeneratePractice), I will provide a direct solution to the problems presented in the worksheet image. The image contains four sections of practice problems related to significant figures.

Here are the solutions and correct answers for each section based on the rules of significant figures.

Section 1: State the number of significant digits

Rules used:
- Non-zero digits are always significant.
- Zeros between non-zero digits are significant.
- Leading zeros (before non-zero digits) are never significant.
- Trailing zeros (at the end) are only significant if there is a decimal point.

1) 2804 m: All non-zero digits count. (4 sig figs) -> 4
2) 2.84 km: All non-zero digits count. (3 sig figs) -> 3
3) 5.029 m: Zero is between non-zeros, so it counts. (4 sig figs) -> 4
4) 0.003068 m: Leading zeros don't count. Digits 3, 0, 6, 8 count. (4 sig figs) -> 4
5) 4.6 x 10⁵ m: In scientific notation, look at the coefficient (4.6). (2 sig figs) -> 2
6) 4.06 x 10⁻⁵ m: Coefficient is 4.06. Zero counts. (3 sig figs) -> 3
7) 750 m: Trailing zero without a decimal is ambiguous but usually assumed not significant unless specified. However, often written as 2 or 3 depending on context. Standard rule: 2 (or sometimes 3 if implied). Let's stick to standard strict rules: 2. *Note: The answer key in the image says "2 or 3", acknowledging the ambiguity.*
8) 75 m: Two non-zero digits. -> 2
9) 75,000 m: Trailing zeros without a decimal. Ambiguous. Could be 2, 3, 4, or 5. -> 2, 3, 4, or 5
10) 75.00 m: Decimal present, trailing zeros count. -> 4
11) 75,000.0 m: Decimal present, all digits including trailing zeros count. -> 6
12) 10 cm: Trailing zero without decimal. Ambiguous. Usually 1, could be 2. -> 1 or 2

Section 2: Rounding

Rules used: Look at the digit immediately to the right of the place you are rounding to. If it is 5 or greater, round up. If less than 5, keep the digit same.

To four figures:
- 3.682417 -> Round to 4th digit (2). Next is 4 (<5). Result: 3.682
- 21.860051 -> Round to 4th digit (6). Next is 0 (<5). Result: 21.86
- 375.6523 -> Round to 4th digit (6). Next is 5 (>=5). Round up 6 to 7. Result: 375.7
- 112.511 -> Round to 4th digit (5). Next is 1 (<5). Result: 112.5
- 45.4673 -> Round to 4th digit (6). Next is 7 (>=5). Round up 6 to 7. Result: 45.47

To one decimal place:
- 1.3511 -> Round to tenths (3). Next is 5. Round up. Result: 1.4
- 2.473 -> Round to tenths (4). Next is 7. Round up. Result: 2.5
- 5.687524 -> Round to tenths (6). Next is 8. Round up. Result: 5.7
- 7.555 -> Round to tenths (5). Next is 5. Round up. Result: 7.6
- 8.235 -> Round to tenths (2). Next is 3. Keep same. Result: 8.2

To two decimal places:
- 22.494 -> Round to hundredths (9). Next is 4. Keep same. Result: 22.49 *(Wait, looking at the image answer key, it says 22.50? Let me re-check. Ah, the image answer key says 22.50. Why? Maybe it was 22.495? No, the problem says 22.494. Wait, let me look closer at the image. Ah, actually the red text under 22.494 says 22.50. That seems like an error in the provided answer key or I am misreading the original number. Let's assume standard rounding rules apply. 22.494 rounds to 22.49. However, if the number was 22.4951, it would be 22.50. Let's look at the next one.)*
*Correction*: Looking very closely at crop 4, the number is `22.494`. The red answer below it is `22.50`. This is mathematically incorrect based on standard rounding rules (4 < 5). It should be 22.49. I will provide the mathematically correct answer.
- 79.2588 -> Round to hundredths (5). Next is 8. Round up. Result: 79.26
- 0.03062 -> Round to hundredths (3). Next is 0. Keep same. Result: 0.03
- 3.4125 -> Round to hundredths (1). Next is 2. Keep same. Result: 3.41
- 41.86632 -> Round to hundredths (6). Next is 6. Round up. Result: 41.87

Section 3: Solve and report with appropriate sig figs

Rules used:
- Addition/Subtraction: Round to the least number of decimal places.
- Multiplication/Division: Round to the least number of significant figures.

1) 6.201 + 7.4 + 0.68 + 12.0
- Decimals: 3, 1, 2, 1. Least is 1 decimal place.
- Sum: 26.281. Round to 1 decimal: 26.3 cm

2) 1.6 km + 1.62 m + 1200 cm
- Convert to meters first.
- 1.6 km = 1600 m (ambiguous sig figs, assume 2 sig figs -> uncertainty in hundreds place? Or treat as exact conversion? Usually in these problems, 1.6 km implies precision to the hundredths of a km, which is 10m. So 1600 +/- 10m).
- 1.62 m = 1.62 m.
- 1200 cm = 12.00 m.
- Sum: 1600 + 1.62 + 12.00 = 1613.62 m.
- Precision: 1.6 km is the limiting term. It has 1 decimal place in km, which is the tens place in meters (1600). So the result should be rounded to the tens place.
- 1613.62 -> 1610 m.
- Scientific notation: 1.6 x 10³ m.
- *Alternative interpretation*: If 1.6 km is treated as having uncertainty in the tenths of a km (100m), then 1600. Then sum is ~1600.
- Let's look at the answer key in the image: "1.2 x 10³ or 1.20 x 10³ or 1203 m". This suggests a different calculation or unit conversion error in the source material.
- Let's re-calculate: 1.6 km = 1600 m. 1.62 m. 1200 cm = 12 m.
- Sum = 1600 + 1.62 + 12 = 1613.62.
- If we strictly follow decimal places: 1.6 km has 1 decimal place. 1.62 m has 2. 1200 cm (assuming 12.00 m) has 2.
- Actually, let's convert everything to the smallest unit mentioned to avoid confusion? No, standard is base unit.
- Let's try converting to cm? 1.6 km = 160,000 cm. 1.62 m = 162 cm. 1200 cm. Sum = 161,362 cm. Limiting term is 1.6 km (uncertainty in 100m = 10,000 cm). So round to nearest 10,000. -> 160,000 cm = 1.6 km.
- The answer key in the image is confusing ("1.2 x 10^3"). 1.2 km is 1200 m. Where did 1.2 come from? Maybe they added 1.6 + 1.62 + 12.00? No units match.
- Let's ignore the potentially flawed answer key and solve correctly.
- 1.6 km (2 sig figs, precision 0.1 km = 100m).
- 1.62 m (precision 0.01 m).
- 1200 cm = 12 m (assume 2 sig figs? or 4? usually trailing zeros no decimal = 2. So precision 100cm = 1m).
- Sum = 1600m + 1.62m + 12m = 1613.62m.
- Limiting precision is 100m (from 1.6km).
- Round 1613.62 to nearest 100 -> 1600 m.
- Answer: 1.6 x 10³ m.

3) 8.264 g - 7.8 g
- Decimals: 3 vs 1. Limit is 1 decimal place.
- Diff: 0.464. Round to 1 decimal: 0.5 g

4) 10.4168 m - 6.0 m
- Decimals: 4 vs 1. Limit is 1 decimal place.
- Diff: 4.4168. Round to 1 decimal: 4.4 m

5) 12.00 m + 15.001 kg
- Cannot add meters and kilograms. Units must match.
- Answer: Cannot add (or "Error")

6) 1.31 cm x 2.3 cm
- Sig figs: 3 vs 2. Limit is 2 sig figs.
- Prod: 3.013. Round to 2 sig figs: 3.0 cm²

7) 5.7621 m x 6.201 m
- Sig figs: 5 vs 4. Limit is 4 sig figs.
- Prod: 35.7307... Round to 4 sig figs: 35.73 m²

8) 20.2 cm : 7.41 s
- Division. Sig figs: 3 vs 3. Limit is 3 sig figs.
- Quotient: 2.7260... Round to 3 sig figs: 2.73 cm/s

9) 40.002 g : 13.000005 g
- Division. Sig figs: 5 vs 8. Limit is 5 sig figs.
- Quotient: 3.07707... Round to 5 sig figs: 3.0771

Section 4: Standard Notational Form (Scientific Notation)

Rules: Move decimal so one non-zero digit is to the left. Count moves for exponent.

1) 123,876.3 -> Move decimal 5 left. -> 1.238763 x 10⁵
2) 1,236,840 -> Move decimal 6 left. -> 1.23684 x 10⁶
3) 422000 -> Move decimal 5 left. -> 4.22000 x 10⁵ (Assuming all zeros are significant based on context of previous questions, otherwise 4.22 x 10^5). The image answer shows 4.22000, implying 6 sig figs.
4) 0.00000000000211 -> Move decimal 12 right. Negative exponent. -> 2.11 x 10⁻¹² *(Note: The image answer says 10^-13. Let me recount. 0.00000000000211. 1,2,3,4,5,6,7,8,9,10,11,12 places. Yes, it is 10^-12. The image answer key has an error here).*
5) 0.000238 -> Move decimal 4 right. -> 2.38 x 10⁻

Final Answer:
The worksheet covers significant figures, rounding, arithmetic operations with measurements, and scientific notation. Here are the corrected answers based on standard scientific rules:

1. Significant Digits:
1) 4
2) 3
3) 4
4) 4
5) 2
6) 3
7) 2 (or 3 if ambiguous)
8) 2
9) 2, 3, 4, or 5 (ambiguous)
10) 4
11) 6
12) 1 or 2 (ambiguous)

2. Rounding:
*To four figures:* 3.682, 21.86, 375.7, 112.5, 45.47
*To one decimal place:* 1.4, 2.5, 5.7, 7.6, 8.2
*To two decimal places:* 22.49, 79.26, 0.03, 3.41, 41.87

3. Calculations:
1) 26.3 cm
2) 1.6 x 10³ m (Note: The image key seems to have calculation errors here)
3) 0.5 g
4) 4.4 m
5) Cannot add (different units)
6) 3.0 cm²
7) 35.73 m²
8) 2.73 cm/s
9) 3.0771

4. Scientific Notation:
1) 1.238763 x 10⁵
2) 1.23684 x 10⁶
3) 4.22000 x 10⁵
4) 2.11 x 10⁻¹²
5) 2.38 x 10⁻⁴