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Let's solve each of the questions step-by-step:
### 9. Round off each of the following numbers to the indicated number of significant figures:
a. 0.45555 (4 sig figs):
- Original number: 0.45555
- Considering the first 4 significant figures: 0.4555
- Rounding off the 5th figure (which is 5), we round up the fourth figure.
Result: 0.4556
b. 292.49 (3 sig figs):
- Original number: 292.49
- Considering the first 3 significant figures: 292
- The number after the third figure is 0.49; since the first figure after the cutoff is '4', we round down.
Result: 292
c. 17.0051 (4 sig figs):
- Original number: 17.0051
- Considering the first 4 significant figures: 17.00
- The number after the fourth figure is 51; since the first figure after the cutoff is '5', we round up.
Result: 17.01
d. 432.969 (5 sig figs):
- Original number: 432.969
- Considering the first 5 significant figures: 432.96
- The number after the fifth figure is 9; since the first figure after the cutoff is '9', we round up.
Result: 432.97
### 10. Evaluate each of the following, and write the answer to the appropriate number of significant figures:
a. [tex]\(1.094 \times 10^5 - 1.073 \times 10^4\)[/tex]:
- Convert both numbers to the same exponent: [tex]\(1.073 \times 10^4\)[/tex] becomes [tex]\(0.1073 \times 10^5\)[/tex]
- Subtract: [tex]\(1.094 \times 10^5 - 0.1073 \times 10^5 = 0.9867 \times 10^5\)[/tex].
- The initial precision is 4 significant figures.
Result: [tex]\(0.9867 \times 10^5\)[/tex] or [tex]\(98,670\)[/tex]
b. [tex]\(\left(3.923 \times 10^2 \text{ cm}\right)\left(2.94 \text{ cm}\right)\left(4.093 \times 10^{-3} \text{ cm}\right)\)[/tex]:
- Multiply the constants: [tex]\(3.923 \times 2.94 \times 4.093 = 47.20855\)[/tex].
- Combine exponents: [tex]\(10^2 \times 10^{-3} = 10^{-1}\)[/tex].
- Total multiplication: [tex]\(47.20855 \times 10^{-1} = 4.720855 \text{ cm}^3\)[/tex] (Rounded to 3 significant figures as per the lease precise figure [tex]\(2.94\)[/tex]).
Result: [tex]\(4.72 \text{ cm}^3\)[/tex]
c. [tex]\(\left(2.932 \times 10^4 \text{ m}\right)[2.404 \times 10^2 \text{ m} + 1.32 \times 10^1 \text{ m}]\)[/tex]:
- Perform addition inside brackets: [tex]\(2.404 \times 10^2 + 1.32 \times 10^1 = 2.536 \times 10^2 \text{ m}\)[/tex].
- Multiply: [tex]\(2.932 \times 10^4 \times 2.536 \times 10^2 = 7.433952 \times 10^6 \text{ m}^2\)[/tex].
- Round to the appropriate precision based on the least precise figure (3 sig figs from [tex]\(1.32 \times 10^1\)[/tex]).
Result: [tex]\(7.43 \times 10^6 \text{ m}^2\)[/tex]
d. [tex]\(\left[2.34 \times 10^2 \text{ g} + 4.443 \times 10^{-1} \text{ g}\right] / (0.0323 \text{ mL})\)[/tex]:
- Perform addition inside brackets: [tex]\(2.34 \times 10^2 + 0.4443 = 234.4443 \text{ g}\)[/tex].
- Divide: [tex]\(234.4443 / 0.0323 = 7250.45944\)[/tex].
- Round to the appropriate precision based on the least precise figure (3 sig figs from [tex]\(0.0323\)[/tex]).
Result: [tex]\(7.25 \times 10^3 \text{ g/mL}\)[/tex]
### 1. How many dozen eggs are there in 48 eggs?
To find out how many dozen (sets of 12) eggs are in 48 eggs:
- Divide the total number of eggs by the number of eggs in one dozen:
[tex]\[ \text{Number of dozens} = \frac{48 \text{ eggs}}{12 \text{ eggs/dozen}} = 4 \text{ dozens} \][/tex]
Result: 4 dozens
### 9. Round off each of the following numbers to the indicated number of significant figures:
a. 0.45555 (4 sig figs):
- Original number: 0.45555
- Considering the first 4 significant figures: 0.4555
- Rounding off the 5th figure (which is 5), we round up the fourth figure.
Result: 0.4556
b. 292.49 (3 sig figs):
- Original number: 292.49
- Considering the first 3 significant figures: 292
- The number after the third figure is 0.49; since the first figure after the cutoff is '4', we round down.
Result: 292
c. 17.0051 (4 sig figs):
- Original number: 17.0051
- Considering the first 4 significant figures: 17.00
- The number after the fourth figure is 51; since the first figure after the cutoff is '5', we round up.
Result: 17.01
d. 432.969 (5 sig figs):
- Original number: 432.969
- Considering the first 5 significant figures: 432.96
- The number after the fifth figure is 9; since the first figure after the cutoff is '9', we round up.
Result: 432.97
### 10. Evaluate each of the following, and write the answer to the appropriate number of significant figures:
a. [tex]\(1.094 \times 10^5 - 1.073 \times 10^4\)[/tex]:
- Convert both numbers to the same exponent: [tex]\(1.073 \times 10^4\)[/tex] becomes [tex]\(0.1073 \times 10^5\)[/tex]
- Subtract: [tex]\(1.094 \times 10^5 - 0.1073 \times 10^5 = 0.9867 \times 10^5\)[/tex].
- The initial precision is 4 significant figures.
Result: [tex]\(0.9867 \times 10^5\)[/tex] or [tex]\(98,670\)[/tex]
b. [tex]\(\left(3.923 \times 10^2 \text{ cm}\right)\left(2.94 \text{ cm}\right)\left(4.093 \times 10^{-3} \text{ cm}\right)\)[/tex]:
- Multiply the constants: [tex]\(3.923 \times 2.94 \times 4.093 = 47.20855\)[/tex].
- Combine exponents: [tex]\(10^2 \times 10^{-3} = 10^{-1}\)[/tex].
- Total multiplication: [tex]\(47.20855 \times 10^{-1} = 4.720855 \text{ cm}^3\)[/tex] (Rounded to 3 significant figures as per the lease precise figure [tex]\(2.94\)[/tex]).
Result: [tex]\(4.72 \text{ cm}^3\)[/tex]
c. [tex]\(\left(2.932 \times 10^4 \text{ m}\right)[2.404 \times 10^2 \text{ m} + 1.32 \times 10^1 \text{ m}]\)[/tex]:
- Perform addition inside brackets: [tex]\(2.404 \times 10^2 + 1.32 \times 10^1 = 2.536 \times 10^2 \text{ m}\)[/tex].
- Multiply: [tex]\(2.932 \times 10^4 \times 2.536 \times 10^2 = 7.433952 \times 10^6 \text{ m}^2\)[/tex].
- Round to the appropriate precision based on the least precise figure (3 sig figs from [tex]\(1.32 \times 10^1\)[/tex]).
Result: [tex]\(7.43 \times 10^6 \text{ m}^2\)[/tex]
d. [tex]\(\left[2.34 \times 10^2 \text{ g} + 4.443 \times 10^{-1} \text{ g}\right] / (0.0323 \text{ mL})\)[/tex]:
- Perform addition inside brackets: [tex]\(2.34 \times 10^2 + 0.4443 = 234.4443 \text{ g}\)[/tex].
- Divide: [tex]\(234.4443 / 0.0323 = 7250.45944\)[/tex].
- Round to the appropriate precision based on the least precise figure (3 sig figs from [tex]\(0.0323\)[/tex]).
Result: [tex]\(7.25 \times 10^3 \text{ g/mL}\)[/tex]
### 1. How many dozen eggs are there in 48 eggs?
To find out how many dozen (sets of 12) eggs are in 48 eggs:
- Divide the total number of eggs by the number of eggs in one dozen:
[tex]\[ \text{Number of dozens} = \frac{48 \text{ eggs}}{12 \text{ eggs/dozen}} = 4 \text{ dozens} \][/tex]
Result: 4 dozens
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