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?

Question

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:

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