Answered

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?


A$\frac{4 cm}{1 \text { year }} \times \frac{10 mm}{1 cm} \times \frac{1 \text { year }}{365 \text { days }}$
B$\frac{4 cm}{1 \text { year }} \times \frac{1 mm}{10 cm} \times \frac{1 \text { year }}{365 \text { days }}$
C$\frac{4 cm}{1 \text { year }} \times \frac{1 cm}{10 mm} \times \frac{365 \text { day } s }{1 \text { year }}$
D$\frac{4 cm}{1 \text { year }} \times \frac{10 mm}{1 cm} \times \frac{365 \text { days }}{1 \text { year }}$

Verified Answer :

  • Start with the rate of \(4 \text{ cm/year}\).
  • Convert centimeters to millimeters using the ratio \(\frac{10 \text{ mm}}{1 \text{ cm}}\).
  • Convert years to days using the ratio \(\frac{1 \text{ year}}{365 \text{ days}}\).
  • The resulting expression is \(\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}}\), which is the first option \(\boxed{\frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}}}\).

Explanation

  1. Analyzing the Conversion Requirements

To convert the rate of erosion from centimeters per year to millimeters per day, we need to apply appropriate conversion factors. The initial rate is given as \(4 \text{ cm/year}\). Our target is to express this in units of \(\text{mm/day}\). We know the following relationships:

  1. \(1 \text{ cm} = 10 \text{ mm}\)
  2. \(1 \text{ year} = 365 \text{ days}\)
  1. Developing the Dimensional Analysis Expression

We start with the original rate:
\[ \frac{4 \text{ cm}}{1 \text{ year}} \]
To eliminate \(\text{cm}\) and introduce \(\text{mm}\), we multiply by the ratio \(\frac{10 \text{ mm}}{1 \text{ cm}}\).
To eliminate \(\text{year}\) and introduce \(\text{days}\), we multiply by the ratio \(\frac{1 \text{ year}}{365 \text{ days}}\).
Combining these, we get:
\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}} \]
Checking the units: \(\frac{\text{cm}}{\text{year}} \times \frac{\text{mm}}{\text{cm}} \times \frac{\text{year}}{\text{day}} = \frac{\text{mm}}{\text{day}}\). The units cancel correctly, leaving \(\text{mm/day}\).

  1. Selecting the Correct Expression

Comparing our derived expression with the provided options, we see it matches the first option:
\[ \frac{4 \text{ cm}}{1 \text{ year}} \times \frac{10 \text{ mm}}{1 \text{ cm}} \times \frac{1 \text{ year}}{365 \text{ days}} \]
The other options either use the wrong conversion ratio (e.g., \(1 \text{ mm}/10 \text{ cm}\)), place units incorrectly (e.g., \(\text{days}\) in the numerator), or invert the ratio between years and days. Therefore, the first option is the correct one.

Examples

Dimensional analysis is a fundamental skill in science and engineering. For example, if a car travels at a certain speed in miles per hour, an engineer might need to convert that speed to meters per second to perform physics calculations for crash testing. By setting up the conversion factors such that unwanted units cancel out, you ensure the final value is accurate and correctly labeled. This same logic applies to everything from calculating dosage in medicine to optimizing industrial processes.

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