Answer :
Let's tackle this problem by examining each given expression to determine which one correctly converts the erosion rate from centimeters per year to millimeters per day.
### Given Erosion Rate
The initial erosion rate is given as [tex]\( 4 \text{ centimeters per year} \)[/tex].
### Conversion Factors
- We need to convert centimeters to millimeters. Since [tex]\( 1 \text{ centimeter} = 10 \text{ millimeters} \)[/tex], the conversion factor is [tex]\( \frac{10 \text{ mm}}{1 \text{ cm}} \)[/tex].
- We also need to convert years to days. Since [tex]\( 1 \text{ year} = 365 \text{ days} \)[/tex], the conversion factor is [tex]\( \frac{1 \text{ year}}{365 \text{ days}} \)[/tex].
Now let's evaluate each of the given expressions step-by-step.
### Option 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, convert centimeters to millimeters: [tex]\( 4 \text{ cm} \times \frac{10 \text{ mm}}{1 \text{ cm}} = 40 \text{ mm} \)[/tex].
- Then convert years to days: [tex]\( 40 \text{ mm per year} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{40 \text{ mm}}{365 \text{ days}} \approx 0.1095890410958904 \text{ mm per day} \)[/tex].
### Option 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, convert centimeters to millimeters incorrectly: [tex]\( 4 \text{ cm} \times \frac{1 \text{ mm}}{10 \text{ cm}} = 0.4 \text{ mm} \)[/tex].
- Then convert years to days: [tex]\( 0.4 \text{ mm per year} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{0.4 \text{ mm}}{365 \text{ days}} \approx 0.0010958904109589042 \text{ mm per day} \)[/tex].
### Option 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, the conversion factor for millimeters and centimeters is inverted: [tex]\( 4 \text{ cm} \times \frac{1 \text{ cm}}{10 \text{ mm}} = 0.4 \text{ cm} \)[/tex].
- Then multiply by the number of days per year [tex]\( 0.4 \text{ mm} \times 365 \approx 146 \text{ mm per day} \)[/tex], which is logically incorrect because it increases the rate significantly.
### Option 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, convert centimeters to millimeters [tex]\( 4 \text{ cm} \times \frac{10 \text{ mm}}{1 \text{ cm}} = 40 \text{ mm} \)[/tex].
- Then multiply by the number of days per year [tex]\( 40 \text{ mm} \times 365 = 14600 \text{ mm per day} \)[/tex], which again significantly increases the rate, hence incorrect.
### Conclusion
The expression that gives the correct unit and the 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 results in an erosion rate of approximately [tex]\( 0.1096 \text{ mm per day} \)[/tex], thus confirming that the correct option is Option 1.
### Given Erosion Rate
The initial erosion rate is given as [tex]\( 4 \text{ centimeters per year} \)[/tex].
### Conversion Factors
- We need to convert centimeters to millimeters. Since [tex]\( 1 \text{ centimeter} = 10 \text{ millimeters} \)[/tex], the conversion factor is [tex]\( \frac{10 \text{ mm}}{1 \text{ cm}} \)[/tex].
- We also need to convert years to days. Since [tex]\( 1 \text{ year} = 365 \text{ days} \)[/tex], the conversion factor is [tex]\( \frac{1 \text{ year}}{365 \text{ days}} \)[/tex].
Now let's evaluate each of the given expressions step-by-step.
### Option 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, convert centimeters to millimeters: [tex]\( 4 \text{ cm} \times \frac{10 \text{ mm}}{1 \text{ cm}} = 40 \text{ mm} \)[/tex].
- Then convert years to days: [tex]\( 40 \text{ mm per year} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{40 \text{ mm}}{365 \text{ days}} \approx 0.1095890410958904 \text{ mm per day} \)[/tex].
### Option 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, convert centimeters to millimeters incorrectly: [tex]\( 4 \text{ cm} \times \frac{1 \text{ mm}}{10 \text{ cm}} = 0.4 \text{ mm} \)[/tex].
- Then convert years to days: [tex]\( 0.4 \text{ mm per year} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{0.4 \text{ mm}}{365 \text{ days}} \approx 0.0010958904109589042 \text{ mm per day} \)[/tex].
### Option 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, the conversion factor for millimeters and centimeters is inverted: [tex]\( 4 \text{ cm} \times \frac{1 \text{ cm}}{10 \text{ mm}} = 0.4 \text{ cm} \)[/tex].
- Then multiply by the number of days per year [tex]\( 0.4 \text{ mm} \times 365 \approx 146 \text{ mm per day} \)[/tex], which is logically incorrect because it increases the rate significantly.
### Option 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, convert centimeters to millimeters [tex]\( 4 \text{ cm} \times \frac{10 \text{ mm}}{1 \text{ cm}} = 40 \text{ mm} \)[/tex].
- Then multiply by the number of days per year [tex]\( 40 \text{ mm} \times 365 = 14600 \text{ mm per day} \)[/tex], which again significantly increases the rate, hence incorrect.
### Conclusion
The expression that gives the correct unit and the 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 results in an erosion rate of approximately [tex]\( 0.1096 \text{ mm per day} \)[/tex], thus confirming that the correct option is Option 1.
