Question

A particular beach is eroding at a rate of 4 centimeters per year. A realtor converts this rate to millimeters per day. Which expression, when evaluated, results in the correct units and numerical value?

StartFraction 4 centimeters Over 1 year EndFraction times StartFraction 10 millimeters Over 1 centimeter EndFraction times StartFraction 1 year Over 365 days EndFraction
StartFraction 4 centimeters Over 1 year EndFraction times StartFraction 1 millimeters Over 10 centimeter EndFraction times StartFraction 1 year Over 365 days EndFraction
StartFraction 4 centimeters Over 1 year EndFraction times StartFraction 1 centimeter Over 10 millimeters EndFraction times StartFraction 365 days Over 1 year EndFraction
StartFraction 4 centimeters Over 1 year EndFraction times StartFraction 10 millimeters Over 1 centimeter EndFraction times StartFraction 365 days Over 1 year EndFraction

Answer 1

To convert the rate of beach erosion from centimeters per year to millimeters per day, we need to set up the right conversion factors step by step.

1. **Convert centimeters to millimeters**: There are 10 millimeters in 1 centimeter. Therefore, to convert centimeters to millimeters, we multiply by \( 10 \: \text{mm/cm} \).

2. **Convert years to days**: There are 365 days in 1 year. To convert years to days, we will multiply by \( \frac{1 \: \text{year}}{365 \: \text{days}} \).

Now, let's analyze the first option:

\[
\frac{4 \: \text{cm}}{1 \: \text{year}} \times \frac{10 \: \text{mm}}{1 \: \text{cm}} \times \frac{1 \: \text{year}}{365 \: \text{days}}
\]

- Here \( \frac{10 \: \text{mm}}{1 \: \text{cm}} \) converts centimeters to millimeters,
- \( \frac{1 \: \text{year}}{365 \: \text{days}} \) converts years to days,
- The units of years cancel out, and centimeters cancel out, resulting in millimeters per day.

Calculating the numerical value:

\[
\frac{4 \: \text{cm}}{1 \: \text{year}} \times \frac{10 \: \text{mm}}{1 \: \text{cm}} \times \frac{1 \: \text{year}}{365 \: \text{days}} = \frac{4 \times 10}{365} \: \text{mm/day}
\]

This gives us the correct result in units of millimeters per day.

Now, let's check if the other options could work:

- **Second option**:
\[
\frac{4 \: \text{cm}}{1 \: \text{year}} \times \frac{1 \: \text{mm}}{10 \: \text{cm}} \times \frac{1 \: \text{year}}{365 \: \text{days}}
\]
This will produce the wrong result, as \( \frac{1 \: \text{mm}}{10 \: \text{cm}} \) would provide the wrong conversion, resulting in mm/year instead of mm/day.

- **Third option**:
\[
\frac{4 \: \text{cm}}{1 \: \text{year}} \times \frac{1 \: \text{cm}}{10 \: \text{mm}} \times \frac{365 \: \text{days}}{1 \: \text{year}}
\]
This option also produces the wrong units because \( \frac{1 \: \text{cm}}{10 \: \text{mm}} \) indicates a conversion that is flipped.

- **Fourth option**:
\[
\frac{4 \: \text{cm}}{1 \: \text{year}} \times \frac{10 \: \text{mm}}{1 \: \text{cm}} \times \frac{365 \: \text{days}}{1 \: \text{year}}
\]
This produces mm/year instead of mm/day, which is again incorrect.

Therefore, the correct expression that results in the desired units of millimeters per day with the correct numerical value is:

\[
\frac{4 \: \text{cm}}{1 \: \text{year}} \times \frac{10 \: \text{mm}}{1 \: \text{cm}} \times \frac{1 \: \text{year}}{365 \: \text{days}}
\]