Dimensional Analysis worksheet with a cartoon character pondering a math problem involving unit conversion.

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Model 1: Dimensional Analysis

#### Cartoon Description (Likely Content):
The cartoon shows a scientist or student at a lab bench. There's a drawing of a beaker labeled "500 mL" and a thought bubble saying:
> “600 × 3 = 1800... so 1800 mL?”

This suggests a misunderstanding of units — the student is multiplying numbers without considering the units, which is a classic example of incorrect dimensional analysis.

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Question 1:

> In the drawing of the student's thought, what does "1800" represent?

Answer:
"1800" represents the result of multiplying 600 by 3, but without proper unit consideration. It appears the student is trying to calculate something involving volume (e.g., 500 mL), but incorrectly assumes that multiplying numbers gives the correct answer regardless of units. The number 1800 is dimensionally incorrect because it doesn’t account for the actual physical meaning of the quantities involved.

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Question 2:

> In the student’s thoughts, which number is a conversion factor?

Answer:
The number 3 is likely the conversion factor, assuming the student is trying to convert between units (e.g., from mL to L, or perhaps scaling up). But since 600 × 3 = 1800, and the original volume is 500 mL, this suggests confusion. A correct conversion might involve something like:
- 1 L = 1000 mL → so 500 mL = 0.5 L

But if the student says 600 × 3 = 1800, maybe they're misapplying a factor (e.g., thinking 600 mL × 3 = 1800 mL, but that’s not relevant to 500 mL).

So, 3 is the conversion factor, but it’s being applied incorrectly.

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Question 3:

> Using the graphically correct sentence, describe a typical dimensional analysis of a conversion factor. In other words, how can you recognize a conversion factor when you see one?

Answer:
A conversion factor is a ratio of two equivalent quantities expressed in different units. It allows you to convert from one unit to another without changing the actual value.

For example:
- \( \frac{1000\ \text{mL}}{1\ \text{L}} \) or \( \frac{1\ \text{L}}{1000\ \text{mL}} \)

These are dimensionally consistent because they equal 1 (they’re just different ways of expressing the same amount).

To recognize a conversion factor:
- It must have the same physical quantity in both numerator and denominator.
- It must be equal to 1 (in terms of magnitude).
- It allows you to cancel units correctly.

Example:
Convert 500 mL to liters:
\[
500\ \text{mL} \times \frac{1\ \text{L}}{1000\ \text{mL}} = 0.5\ \text{L}
\]

The mL units cancel, leaving L.

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Question 4:

> If you were to apply the problem, you could use dimensional analysis to find the number of seconds in 2 days. Show your work.

Answer:
We want to convert 2 days into seconds.

Use step-by-step conversion factors:

1. \( 1\ \text{day} = 24\ \text{hours} \)
2. \( 1\ \text{hour} = 60\ \text{minutes} \)
3. \( 1\ \text{minute} = 60\ \text{seconds} \)

Now set up the dimensional analysis:

\[
2\ \text{days} \times \frac{24\ \text{hr}}{1\ \text{day}} \times \frac{60\ \text{min}}{1\ \text{hr}} \times \frac{60\ \text{sec}}{1\ \text{min}}
\]

Cancel units step by step:
- days cancel
- hr cancel
- min cancel

Left with:
\[
2 \times 24 \times 60 \times 60 = 172,800\ \text{seconds}
\]

✔ So, 2 days = 172,800 seconds

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Final Answer Summary:

1. "1800" is an incorrect numerical result due to improper unit handling.
2. The number 3 is the intended conversion factor (but used incorrectly).
3. A conversion factor is a ratio of equivalent quantities with different units (e.g., 1000 mL / 1 L), used to cancel units and convert measurements.
4. 2 days = 172,800 seconds (via dimensional analysis).

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