Answer :
To determine which expression correctly converts the erosion rate from centimeters per year to millimeters per day, let's analyze each option step by step.
### Conversion Factors:
- \(1 \text{ cm} = 10 \text{ mm}\)
- \(1 \text{ year} = 365 \text{ days}\)
### Expression (A):
[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]
Step-by-Step Calculation:
1. Convert centimeters to millimeters:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} = \frac{40 \text{ mm}}{1 \text{ year}} \][/tex]
2. Convert years to days:
[tex]\[ \frac{40 \text{ mm}}{1 \text{ year}} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{40 \text{ mm}}{365 \text{ days}} \][/tex]
3. Simplify the fraction:
[tex]\[ \frac{40 \text{ mm}}{365 \text{ days}} \approx 0.1095890410958904 \text{ mm per day} \][/tex]
This conversion yields a value of approximately \(0.10959 \text{ mm per day}\), which has the correct numerical magnitude and units.
### Expression (B):
[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]
Step-by-Step Calculation:
1. Convert centimeters to millimeters incorrectly:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ mm}}{10 \text{ cm}} = \frac{4}{10} \text{ mm per year} = 0.4 \text{ mm per year} \][/tex]
2. Convert years to days:
[tex]\[ \frac{0.4 \text{ mm}}{1 \text{ year}} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{0.4 \text{ mm}}{365 \text{ days}} \approx 0.0010958904109589042 \text{ mm per day} \][/tex]
This conversion yields a value of approximately \(0.001095 \text{ mm per day}\), which is significantly smaller than expected and incorrect.
### Expression (C):
[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]
Step-by-Step Calculation:
1. Incorrectly convert centimeters to millimeters:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ cm}}{10 \text{ mm}} = \frac{4}{10} \text{ cm} \times \text{ mm per year} \][/tex]
2. Convert years to days erroneously:
[tex]\[ \frac{0.4 \text{ cm}}{10 \text{ mm per year}} \times \frac{365 \text{ days}}{1 \text{ year}} = 0.4 \text{ cm} \times 365 \text{ mm per day} = 146 \text{ mm per day} \][/tex]
This conversion results in \(146 \text{ mm per day}\), which is too high and incorrect.
### Expression (D):
[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]
Step-by-Step Calculation:
1. Convert centimeters to millimeters:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} = \frac{40 \text{ mm per year}} \][/tex]
2. Incorrectly convert years to days:
[tex]\[ \frac{40 \text{ mm}}{1 \text{ year}} \times \frac{365 \text{ days}}{1 \text{ year}} = 40 \text{ mm} \times 365 \text{ days per year} = 14600 \text{ mm per day} \][/tex]
This results in \(14600 \text{ mm per day}\), which is excessively high and incorrect.
### Conclusion:
The correct expression that accurately converts the erosion rate from centimeters per year to millimeters per day, both in terms of correct units and numerical value, is:
[tex]\[ \boxed{\frac{4 \text{ cm }}{1 \text{ year }} \times \frac{10 \text{ mm }}{1 \text{ cm }} \times \frac{1 \text{ year }}{365 \text{ days }}} \][/tex]
Which evaluates to approximately [tex]\(0.1095890410958904 \text{ mm per day}\)[/tex].
### Conversion Factors:
- \(1 \text{ cm} = 10 \text{ mm}\)
- \(1 \text{ year} = 365 \text{ days}\)
### Expression (A):
[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]
Step-by-Step Calculation:
1. Convert centimeters to millimeters:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} = \frac{40 \text{ mm}}{1 \text{ year}} \][/tex]
2. Convert years to days:
[tex]\[ \frac{40 \text{ mm}}{1 \text{ year}} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{40 \text{ mm}}{365 \text{ days}} \][/tex]
3. Simplify the fraction:
[tex]\[ \frac{40 \text{ mm}}{365 \text{ days}} \approx 0.1095890410958904 \text{ mm per day} \][/tex]
This conversion yields a value of approximately \(0.10959 \text{ mm per day}\), which has the correct numerical magnitude and units.
### Expression (B):
[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]
Step-by-Step Calculation:
1. Convert centimeters to millimeters incorrectly:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ mm}}{10 \text{ cm}} = \frac{4}{10} \text{ mm per year} = 0.4 \text{ mm per year} \][/tex]
2. Convert years to days:
[tex]\[ \frac{0.4 \text{ mm}}{1 \text{ year}} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{0.4 \text{ mm}}{365 \text{ days}} \approx 0.0010958904109589042 \text{ mm per day} \][/tex]
This conversion yields a value of approximately \(0.001095 \text{ mm per day}\), which is significantly smaller than expected and incorrect.
### Expression (C):
[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]
Step-by-Step Calculation:
1. Incorrectly convert centimeters to millimeters:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{1 \text{ cm}}{10 \text{ mm}} = \frac{4}{10} \text{ cm} \times \text{ mm per year} \][/tex]
2. Convert years to days erroneously:
[tex]\[ \frac{0.4 \text{ cm}}{10 \text{ mm per year}} \times \frac{365 \text{ days}}{1 \text{ year}} = 0.4 \text{ cm} \times 365 \text{ mm per day} = 146 \text{ mm per day} \][/tex]
This conversion results in \(146 \text{ mm per day}\), which is too high and incorrect.
### Expression (D):
[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]
Step-by-Step Calculation:
1. Convert centimeters to millimeters:
[tex]\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} = \frac{40 \text{ mm per year}} \][/tex]
2. Incorrectly convert years to days:
[tex]\[ \frac{40 \text{ mm}}{1 \text{ year}} \times \frac{365 \text{ days}}{1 \text{ year}} = 40 \text{ mm} \times 365 \text{ days per year} = 14600 \text{ mm per day} \][/tex]
This results in \(14600 \text{ mm per day}\), which is excessively high and incorrect.
### Conclusion:
The correct expression that accurately converts the erosion rate from centimeters per year to millimeters per day, both in terms of correct units and numerical value, is:
[tex]\[ \boxed{\frac{4 \text{ cm }}{1 \text{ year }} \times \frac{10 \text{ mm }}{1 \text{ cm }} \times \frac{1 \text{ year }}{365 \text{ days }}} \][/tex]
Which evaluates to approximately [tex]\(0.1095890410958904 \text{ mm per day}\)[/tex].
