Solve the following for x, showing the correct number of significant figures in your answers: 0.0100 * 48.2 = 2.4 * 10^x = (x * log(10.5)) 0.00900 * 20.8 Y – 439 Xo, (o log(0.0660 / 9.12) = 0.468 log(295) = (0.05321 * 8.7) 0.488 = (0.003771 * 15.2) 7 * (0.0021 * 72.5) = 0.0509 0.00612 XFoywl 4.60×10^4 = 8.80×10^8.314 * 10^3.75×10^6 = 82.9 * 625 * 305) * 8.314 * 10^623
The Correct Answer and Explanation is:
Let’s tackle each expression, calculate the correct value, and round it properly based on significant figures.
1. 0.0100 × 48.2
- Calculation:
0.0100×48.2=0.4820.0100 × 48.2 = 0.4820.0100×48.2=0.482 - Significant figures: 0.0100 has 3 sig. figs, 48.2 has 3 sig. figs → answer = 0.482 (3 sig. figs)
2. 2.4 × 10^x = (x × log(10.5))
This is an equation with variables. Rearranged:2.4×10x=x×log(10.5)2.4 × 10^x = x × \log(10.5)2.4×10x=x×log(10.5)
Take logs:log(2.4×10x)=log(x×log(10.5))\log(2.4 × 10^x) = \log(x × \log(10.5))log(2.4×10x)=log(x×log(10.5))
This becomes:log(2.4)+x=log(x)+log(log(10.5))\log(2.4) + x = \log(x) + \log(\log(10.5))log(2.4)+x=log(x)+log(log(10.5))
This is transcendental and requires numerical solving. Let’s simplify:
- log(2.4)≈0.3802\log(2.4) ≈ 0.3802log(2.4)≈0.3802
- log(10.5)≈1.021\log(10.5) ≈ 1.021log(10.5)≈1.021
- log(log(10.5))≈log(1.021)≈0.009\log(\log(10.5)) ≈ \log(1.021) ≈ 0.009log(log(10.5))≈log(1.021)≈0.009
Try trial-and-error or numerical methods:
Let’s assume x=1x = 1x=1:
LHS: 2.4×101=242.4 × 10^1 = 242.4×101=24
RHS: 1×log(10.5)=1.0211 × \log(10.5) = 1.0211×log(10.5)=1.021 → not equal
Try x=15x = 15x=15:
LHS: 2.4×1015=2.4e152.4 × 10^{15} = 2.4e152.4×1015=2.4e15
RHS: 15×1.021=15.31515 × 1.021 = 15.31515×1.021=15.315
Very far. So this is likely symbolic only, or not solvable without context.
3. 0.00900 × 20.8
- 0.00900×20.8=0.18720.00900 × 20.8 = 0.18720.00900×20.8=0.1872
- Least sig. figs: 0.00900 → 3 sig. figs
- Final: 0.187 (3 sig. figs)
4. log(0.0660 / 9.12)
- Division: 0.0660/9.12=0.00723680.0660 / 9.12 = 0.00723680.0660/9.12=0.0072368
- Log: log(0.0072368)≈−2.140\log(0.0072368) ≈ -2.140log(0.0072368)≈−2.140
- Sig. figs in log: 3 sig. figs → 3 decimal places
- Final: −2.140
5. 0.468 × log(295)
- log(295)≈2.470\log(295) ≈ 2.470log(295)≈2.470
- Multiply: 0.468×2.470=1.1560.468 × 2.470 = 1.1560.468×2.470=1.156
- Sig. figs: both 3 → 1.16 (3 sig. figs)
6. 0.05321 × 8.7
- 0.05321×8.7=0.4629270.05321 × 8.7 = 0.4629270.05321×8.7=0.462927
- Least sig. figs: 8.7 → 2
- Final: 0.46
7. 0.488 = (0.003771 × 15.2)
- Right side: 0.003771×15.2=0.05731920.003771 × 15.2 = 0.05731920.003771×15.2=0.0573192
- Final: 0.0573 (3 sig. figs)
Not equal to 0.488, so equation is false unless context missing.
8. 7 × (0.0021 × 72.5)
- Inside: 0.0021×72.5=0.152250.0021 × 72.5 = 0.152250.0021×72.5=0.15225
- Multiply: 7×0.15225=1.065757 × 0.15225 = 1.065757×0.15225=1.06575
- Sig. figs: 0.0021 (2), 72.5 (3), 7 (1)
- 1 sig. fig dominates → 1
9. 0.0509 × 0.00612
- 0.0509×0.00612=0.00031150.0509 × 0.00612 = 0.00031150.0509×0.00612=0.0003115
- Sig. figs: 3 × 3 = 3
- Final: 3.12 × 10⁻⁴
10. 4.60×10⁴ = 8.80×10⁸.314 × 10³.75×10⁶
Possibly malformed or missing operator between terms. Also, 8.808.3148.80^{8.314}8.808.314 is ~1.49×10⁸, but the rest is not computable without clear formatting.
11. 82.9 × 625 × 305 × 8.314 × 10⁶²³
That’s:
- Multiply constants:
82.9×625×305×8.314≈1.315×10882.9 × 625 × 305 × 8.314 ≈ 1.315 × 10^882.9×625×305×8.314≈1.315×108 - Final with exponent:
≈1.315×108×10623=1.315×10631≈ 1.315 × 10^8 × 10^{623} = 1.315 × 10^{631}≈1.315×108×10623=1.315×10631 - Final: 1.32 × 10^631 (3 sig. figs)
🔍 Summary of Final Answers (with correct significant figures):
| Expression | Answer |
|---|---|
| 0.0100 × 48.2 | 0.482 |
| 0.00900 × 20.8 | 0.187 |
| log(0.0660 / 9.12) | −2.140 |
| 0.468 × log(295) | 1.16 |
| 0.05321 × 8.7 | 0.46 |
| 0.003771 × 15.2 | 0.0573 |
| 7 × (0.0021 × 72.5) | 1 |
| 0.0509 × 0.00612 | 3.12 × 10⁻⁴ |
| 82.9 × 625 × 305 × 8.314 × 10²³ | 1.32 × 10^631 |
📘 Significant Figures Note:
- Multiplication/Division: Least number of significant figures among terms.
- Logarithms: Number of decimal places in the result equals the number of significant figures in the original number.
- Maintain clarity with scientific notation for very large or small values.
