Answer :
To determine which expression results in the correct units and numerical value for the erosion rate of 4 centimeters per year converted to millimeters per day, let's carefully evaluate each expression step by step:
### Given Data:
- Erosion rate: 4 centimeters per year
- Conversion factors:
- 1 centimeter = 10 millimeters
- 1 year = 365 days
### Expressions to Evaluate:
1. [tex]\(\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}}\)[/tex]
First, multiply [tex]\(\frac{4 \text{ cm}}{1 \text{ year}}\)[/tex] by [tex]\(\frac{10 \text{ mm}}{1 \text{ cm}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} = \frac{4 \times 10 \text{ mm}}{1 \text{ year}} = \frac{40 \text{ mm}}{1 \text{ year}} \][/tex]
Next, multiply by [tex]\(\frac{1 \text{ year}}{365 \text{ days}}\)[/tex]:
[tex]\[ \frac{40 \text{ mm}}{1 \text{ year}} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{40 \text{ mm}}{365 \text{ days}} = \frac{40 \text{ mm}}{365 \text{ days}} \][/tex]
[tex]\[ \frac{40}{365} \approx 0.1095890410958904 \text{ mm/day} \][/tex]
The units are correct (mm/day), and the numerical value is approximately 0.1095890410958904.
2. [tex]\(\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ mm}}{10 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}}\)[/tex]
First, multiply [tex]\(\frac{4 \text{ cm}}{1 \text{ year}}\)[/tex] by [tex]\(\frac{1 \text{ mm}}{10 \text{ cm}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ mm}}{10 \text{ cm}} = \frac{4 \times 1 \text{ mm}}{10 \text{ year}} = \frac{4 \text{ mm}}{10 \text{ year}} = \frac{0.4 \text{ mm}}{1 \text{ year}} \][/tex]
Next, multiply by [tex]\(\frac{1 \text{ year}}{365 \text{ days}}\)[/tex]:
[tex]\[ \frac{0.4 \text{ mm}}{1 \text{ year}} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{0.4 \text{ mm}}{365 \text{ days}} \approx 0.0010958904109589042 \text{ mm/day} \][/tex]
The units are correct (mm/day), but the numerical value is approximately 0.0010958904109589042, which is much smaller than expected.
3. [tex]\(\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ cm}}{10 \text{ mm}} \times \frac{365 \text{ days}}{1 \text{ year}}\)[/tex]
First, multiply [tex]\(\frac{4 \text{ cm}}{1 \text{ year}}\)[/tex] by [tex]\(\frac{1 \text{ cm}}{10 \text{ mm}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ cm}}{10 \text{ mm}} = \frac{4 \text{ cm}}{10 \text{ mm} \times \text{ year}} \][/tex]
Here, the unit conversion is misplaced. Since 1 cm = 10 mm, the correct conversion factor should be [tex]\(\frac{10 \text{ mm}}{1 \text{ cm}}\)[/tex].
Next, multiply by [tex]\(\frac{365 \text{ days}}{1 \text{ year}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{10 \text{ mm} \times \text{ year}} \times \frac{365 \text{ days}}{1 \text{ year}} = \frac{4 \times 365 \text{ days}}{10 \text{ mm}} = 146 \text{ days/mm} \][/tex]
The units are incorrect (days/mm), and the numerical value is 146, which doesn't match the expected result.
4. [tex]\(\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{365 \text{ days}}{1 \text{ year}}\)[/tex]
First, multiply [tex]\(\frac{4 \text{ cm}}{1 \text{ year}}\)[/tex] by [tex]\(\frac{10 \text{ mm}}{1 \text{ cm}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} = \frac{4 \times 10 \text{ mm}}{1 \text{ year}} = \frac{40 \text{ mm}}{1 \text{ year}} \][/tex]
Next, multiply by [tex]\(\frac{365 \text{ days}}{1 \text{ year}}\)[/tex]:
[tex]\[ \frac{40 \text{ mm}}{1 \text{ year}} \times \frac{365 \text{ days}}{1 \text{ year}} = 40 \times 365 \text{ mm/day} = 14600 \text{ mm/day} \][/tex]
The units are incorrect (mm/day), but the numerical value of 14600 is much larger than expected.
### Conclusion
The correct expression matching the expected units (mm/day) and numerical value (approximately 0.1095890410958904 mm/day) is:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}} \][/tex]
Thus, the correct expression is the first one:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}} \][/tex]
### Given Data:
- Erosion rate: 4 centimeters per year
- Conversion factors:
- 1 centimeter = 10 millimeters
- 1 year = 365 days
### Expressions to Evaluate:
1. [tex]\(\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}}\)[/tex]
First, multiply [tex]\(\frac{4 \text{ cm}}{1 \text{ year}}\)[/tex] by [tex]\(\frac{10 \text{ mm}}{1 \text{ cm}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} = \frac{4 \times 10 \text{ mm}}{1 \text{ year}} = \frac{40 \text{ mm}}{1 \text{ year}} \][/tex]
Next, multiply by [tex]\(\frac{1 \text{ year}}{365 \text{ days}}\)[/tex]:
[tex]\[ \frac{40 \text{ mm}}{1 \text{ year}} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{40 \text{ mm}}{365 \text{ days}} = \frac{40 \text{ mm}}{365 \text{ days}} \][/tex]
[tex]\[ \frac{40}{365} \approx 0.1095890410958904 \text{ mm/day} \][/tex]
The units are correct (mm/day), and the numerical value is approximately 0.1095890410958904.
2. [tex]\(\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ mm}}{10 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}}\)[/tex]
First, multiply [tex]\(\frac{4 \text{ cm}}{1 \text{ year}}\)[/tex] by [tex]\(\frac{1 \text{ mm}}{10 \text{ cm}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ mm}}{10 \text{ cm}} = \frac{4 \times 1 \text{ mm}}{10 \text{ year}} = \frac{4 \text{ mm}}{10 \text{ year}} = \frac{0.4 \text{ mm}}{1 \text{ year}} \][/tex]
Next, multiply by [tex]\(\frac{1 \text{ year}}{365 \text{ days}}\)[/tex]:
[tex]\[ \frac{0.4 \text{ mm}}{1 \text{ year}} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{0.4 \text{ mm}}{365 \text{ days}} \approx 0.0010958904109589042 \text{ mm/day} \][/tex]
The units are correct (mm/day), but the numerical value is approximately 0.0010958904109589042, which is much smaller than expected.
3. [tex]\(\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ cm}}{10 \text{ mm}} \times \frac{365 \text{ days}}{1 \text{ year}}\)[/tex]
First, multiply [tex]\(\frac{4 \text{ cm}}{1 \text{ year}}\)[/tex] by [tex]\(\frac{1 \text{ cm}}{10 \text{ mm}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ cm}}{10 \text{ mm}} = \frac{4 \text{ cm}}{10 \text{ mm} \times \text{ year}} \][/tex]
Here, the unit conversion is misplaced. Since 1 cm = 10 mm, the correct conversion factor should be [tex]\(\frac{10 \text{ mm}}{1 \text{ cm}}\)[/tex].
Next, multiply by [tex]\(\frac{365 \text{ days}}{1 \text{ year}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{10 \text{ mm} \times \text{ year}} \times \frac{365 \text{ days}}{1 \text{ year}} = \frac{4 \times 365 \text{ days}}{10 \text{ mm}} = 146 \text{ days/mm} \][/tex]
The units are incorrect (days/mm), and the numerical value is 146, which doesn't match the expected result.
4. [tex]\(\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{365 \text{ days}}{1 \text{ year}}\)[/tex]
First, multiply [tex]\(\frac{4 \text{ cm}}{1 \text{ year}}\)[/tex] by [tex]\(\frac{10 \text{ mm}}{1 \text{ cm}}\)[/tex]:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} = \frac{4 \times 10 \text{ mm}}{1 \text{ year}} = \frac{40 \text{ mm}}{1 \text{ year}} \][/tex]
Next, multiply by [tex]\(\frac{365 \text{ days}}{1 \text{ year}}\)[/tex]:
[tex]\[ \frac{40 \text{ mm}}{1 \text{ year}} \times \frac{365 \text{ days}}{1 \text{ year}} = 40 \times 365 \text{ mm/day} = 14600 \text{ mm/day} \][/tex]
The units are incorrect (mm/day), but the numerical value of 14600 is much larger than expected.
### Conclusion
The correct expression matching the expected units (mm/day) and numerical value (approximately 0.1095890410958904 mm/day) is:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}} \][/tex]
Thus, the correct expression is the first one:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}} \][/tex]
