Answer :
To determine which expression correctly converts an erosion rate of 4 centimeters per year to millimeters per day, we need to use dimensional analysis to ensure the units convert properly and match the desired units.
Let's examine each expression one by one:
### Expression 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]
- Start with \(\frac{4 \, \text{cm}}{1 \, \text{year}}\).
- Convert centimeters to millimeters: \(\frac{10 \, \text{mm}}{1 \, \text{cm}}\).
- Convert years to days: \(\frac{1 \, \text{year}}{365 \, \text{days}}\).
Calculating the units step by step:
[tex]\[ \frac{4 \, \text{cm}}{1 \, \text{year}} \times \frac{10 \, \text{mm}}{1 \, \text{cm}} = \frac{40 \, \text{mm}}{1 \, \text{year}} \][/tex]
Next, 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]
Simplify the fraction:
[tex]\[ \frac{40}{365} \, \text{mm/day} \approx 0.1095890410958904 \, \text{mm/day} \][/tex]
Thus, Expression 1 is correct and evaluates to \(0.1095890410958904 \, \text{mm/day}\).
### Expression 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]
- Convert centimeters to millimeters incorrectly: \(\frac{1 \, \text{mm}}{10 \, \text{cm}}\) which does not make sense as \(1 \, \text{mm}\) = \(0.1 \, \text{cm}\), not the inverse.
- Calculate the final units: \(\frac{4}{3650} \, \text{mm/day}\).
Evaluating:
[tex]\[ \frac{4}{3650} \, \text{mm/day} \approx 0.001095890410958904 \, \text{mm/day} \][/tex]
This is much smaller than the correct answer, so Expression 2 is incorrect.
### Expression 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]
- Convert centimeters to millimeters incorrectly: \(\frac{1 \, \text{cm}}{10 \, \text{mm}}\).
- Calculate the final units: \(\frac{4}{10} \times 365 \, \text{mm/day}\).
Evaluating:
[tex]\[ \frac{1460}{10} = 146 \, \text{mm/day} \][/tex]
This is much larger than the correct answer, so Expression 3 is incorrect.
### Expression 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]
- Convert centimeters to millimeters: \(\frac{10 \, \text{mm}}{1 \, \text{cm}}\).
- Convert years to days incorrectly as \(\frac{365 \, \text{days}}{1 \, \text{year}}\), which expands instead of converting properly.
Evaluating:
[tex]\[ \frac{4 \times 10 \times 365 \, \text{mm}}{1 \times 1 \times 1} = 4 \times 10 \times 365 = 14600 \, \text{mm/day} \][/tex]
This is much larger and incorrect.
### Conclusion:
The correct expression that results in the proper conversion and numerical value 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]
This evaluates correctly to [tex]\(0.1095890410958904 \, \text{mm/day}\)[/tex] and thus matches the correct answer.
Let's examine each expression one by one:
### Expression 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]
- Start with \(\frac{4 \, \text{cm}}{1 \, \text{year}}\).
- Convert centimeters to millimeters: \(\frac{10 \, \text{mm}}{1 \, \text{cm}}\).
- Convert years to days: \(\frac{1 \, \text{year}}{365 \, \text{days}}\).
Calculating the units step by step:
[tex]\[ \frac{4 \, \text{cm}}{1 \, \text{year}} \times \frac{10 \, \text{mm}}{1 \, \text{cm}} = \frac{40 \, \text{mm}}{1 \, \text{year}} \][/tex]
Next, 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]
Simplify the fraction:
[tex]\[ \frac{40}{365} \, \text{mm/day} \approx 0.1095890410958904 \, \text{mm/day} \][/tex]
Thus, Expression 1 is correct and evaluates to \(0.1095890410958904 \, \text{mm/day}\).
### Expression 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]
- Convert centimeters to millimeters incorrectly: \(\frac{1 \, \text{mm}}{10 \, \text{cm}}\) which does not make sense as \(1 \, \text{mm}\) = \(0.1 \, \text{cm}\), not the inverse.
- Calculate the final units: \(\frac{4}{3650} \, \text{mm/day}\).
Evaluating:
[tex]\[ \frac{4}{3650} \, \text{mm/day} \approx 0.001095890410958904 \, \text{mm/day} \][/tex]
This is much smaller than the correct answer, so Expression 2 is incorrect.
### Expression 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]
- Convert centimeters to millimeters incorrectly: \(\frac{1 \, \text{cm}}{10 \, \text{mm}}\).
- Calculate the final units: \(\frac{4}{10} \times 365 \, \text{mm/day}\).
Evaluating:
[tex]\[ \frac{1460}{10} = 146 \, \text{mm/day} \][/tex]
This is much larger than the correct answer, so Expression 3 is incorrect.
### Expression 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]
- Convert centimeters to millimeters: \(\frac{10 \, \text{mm}}{1 \, \text{cm}}\).
- Convert years to days incorrectly as \(\frac{365 \, \text{days}}{1 \, \text{year}}\), which expands instead of converting properly.
Evaluating:
[tex]\[ \frac{4 \times 10 \times 365 \, \text{mm}}{1 \times 1 \times 1} = 4 \times 10 \times 365 = 14600 \, \text{mm/day} \][/tex]
This is much larger and incorrect.
### Conclusion:
The correct expression that results in the proper conversion and numerical value 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]
This evaluates correctly to [tex]\(0.1095890410958904 \, \text{mm/day}\)[/tex] and thus matches the correct answer.
