To convert the erosion rate from centimeters per year to millimeters per day, we need to follow a series of unit conversions. Let's analyze the given options step-by-step with correct unit factors.
Given:
- Rate of erosion: [tex]\( 4 \)[/tex] centimeters per year.
We need to convert this rate into millimeters per day.
Conversion Factors:
1. [tex]\( 1 \text{ cm} = 10 \text{ mm} \)[/tex]
2. [tex]\( 1 \text{ year} = 365 \text{ days} \)[/tex]
Let's look into each option and analyze if they result in the correct units and numerical values:
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]
Simplify this expression:
[tex]\[
4 \text{ cm/year} \times 10 \text{ mm/cm} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{4 \times 10 \text{ mm}}{365 \text{ days}} = \frac{40 \text{ mm}}{365 \text{ days}} \approx 0.1095890410958904 \text{ mm/day}
\][/tex]
This option converts the rate correctly in terms of units and gives a value of approximately [tex]\(0.1095890410958904\)[/tex] mm/day.
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]
Simplify this expression:
[tex]\[
4 \text{ cm/year} \times \frac{1}{10} \text{ mm/cm} \times \frac{1 \text{ year}}{365 \text{ days}} = \frac{4 \times 1 \text{ mm}}{10 \times 365 \text{ days}} = \frac{4 \text{ mm}}{3650 \text{ days}} \approx 0.0010958904109589042 \text{ mm/day}
\][/tex]
This option does not convert the rate correctly and gives an incorrect numerical value.
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]
Simplify this expression:
[tex]\[
4 \text{ cm/year} \times \frac{1 \text{ cm}}{10 \text{ mm}} \times 365 = 4 \times \frac{1}{10} \times 365 \approx 146 \text{ mm/day}
\][/tex]
This option incorrectly applies the conversion factors and gives a value of 146 mm/day.
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]
Simplify this expression:
[tex]\[
4 \text{ cm/year} \times 10 \text{ mm/cm} \times 365 = 4 \times 10 \times 365 = 14600 \text{ mm/day}
\][/tex]
This option yields an incorrect rate and an implausibly high value.
Taking all of this into account, the correct expression that converts the given rate from centimeters per year to millimeters per day with the correct units and value is:
[tex]\[
\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}} \approx 0.1095890410958904 \text{ mm/day}
\][/tex]
So, the correct option is Option 1.
