Dimensional Analysis Worksheet 2020-2024 - Fill and Sign Printable ... - Free Printable

It looks like you've uploaded a worksheet titled "Dimensional Analysis Worksheet - Answers" that includes problems related to unit conversions using dimensional analysis (also known as the factor-label method). Below is a detailed explanation and solution for each problem, based on the content visible in the image.

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🔹 Instructions Recap:

- Use conversion factors to convert units.
- Cancel out units appropriately.
- Ensure all units cancel except the desired one.

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🔹 Part 1: Conversion Factors

#### Problem 1: Convert 4.1984 J to kcal
We are given:
- $ 1 \text{ kcal} = 1000 \text{ cal} $
- $ 1 \text{ cal} = 4.184 \text{ J} $

So,
$$
\text{kcal} = \frac{\text{J}}{4.184 \times 1000}
= \frac{4.1984}{4184} \approx 0.001 \text{ kcal}
$$

But let's do it step-by-step:

$$
4.1984 \text{ J} \times \frac{1 \text{ cal}}{4.184 \text{ J}} \times \frac{1 \text{ kcal}}{1000 \text{ cal}} = ?
$$

Step-by-step:
- First, convert J → cal:
$$
\frac{4.1984}{4.184} \approx 1.0034 \text{ cal}
$$
- Then, convert cal → kcal:
$$
\frac{1.0034}{1000} = 0.0010034 \text{ kcal}
$$

✔ Answer: $ \boxed{0.0010034} $ kcal

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#### Problem 2: Convert 1.000 kJ to cal
Given:
- $ 1 \text{ kJ} = 1000 \text{ J} $
- $ 1 \text{ cal} = 4.184 \text{ J} $

So:
$$
1.000 \text{ kJ} = 1000 \text{ J}
$$
$$
1000 \text{ J} \times \frac{1 \text{ cal}}{4.184 \text{ J}} = \frac{1000}{4.184} \approx 239.0 \text{ cal}
$$

✔ Answer: $ \boxed{239.0} $ cal

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#### Problem 3: Convert 1.000 kJ to kcal
We already know:
- $ 1 \text{ kJ} = 1000 \text{ J} $
- $ 1 \text{ kcal} = 4184 \text{ J} $

So:
$$
1.000 \text{ kJ} = 1000 \text{ J} \times \frac{1 \text{ kcal}}{4184 \text{ J}} = \frac{1000}{4184} \approx 0.2390 \text{ kcal}
$$

✔ Answer: $ \boxed{0.2390} $ kcal

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#### Problem 4: How many cal in 60 Cal?
Note: "Cal" with capital C usually means kilocalorie (kcal), so:
- $ 1 \text{ Cal} = 1 \text{ kcal} = 1000 \text{ cal} $

So:
$$
60 \text{ Cal} = 60 \times 1000 = 60,000 \text{ cal}
$$

✔ Answer: $ \boxed{60,000} $ cal

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#### Problem 5: Convert 10.0 cal to J
Use:
- $ 1 \text{ cal} = 4.184 \text{ J} $

So:
$$
10.0 \text{ cal} \times 4.184 \frac{\text{J}}{\text{cal}} = 41.84 \text{ J}
$$

✔ Answer: $ \boxed{41.84} $ J

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#### Problem 6: Convert 100.0 cal to kJ
First, convert cal → J:
$$
100.0 \text{ cal} \times 4.184 \frac{\text{J}}{\text{cal}} = 418.4 \text{ J}
$$
Then convert J → kJ:
$$
418.4 \text{ J} \div 1000 = 0.4184 \text{ kJ}
$$

✔ Answer: $ \boxed{0.4184} $ kJ

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#### Problem 7: Convert 10.0 J to cal
$$
10.0 \text{ J} \times \frac{1 \text{ cal}}{4.184 \text{ J}} = \frac{10.0}{4.184} \approx 2.39 \text{ cal}
$$

✔ Answer: $ \boxed{2.39} $ cal

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#### Problem 8: Convert 120 Cal to joules
Note: 1 Cal = 1 kcal = 1000 cal
So:
- $ 120 \text{ Cal} = 120 \text{ kcal} = 120,000 \text{ cal} $

Now convert to joules:
$$
120,000 \text{ cal} \times 4.184 \frac{\text{J}}{\text{cal}} = 502,080 \text{ J}
$$

Or in scientific notation:
$$
5.0208 \times 10^5 \text{ J}
$$

✔ Answer: $ \boxed{5.02 \times 10^5} $ J (rounded to 3 significant figures)

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🔹 Bonus Notes:

#### Why does 1 cal ≈ 4.184 J?
This is the definition of the calorie: the amount of energy needed to raise 1 gram of water by 1°C. The exact value was standardized to 4.184 J/cal.

#### Why use dimensional analysis?
It ensures correct unit cancellation and helps avoid errors when converting between systems (e.g., metric vs. imperial or energy units).

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | 0.0010034 kcal |
| 2 | 239.0 cal |
| 3 | 0.2390 kcal |
| 4 | 60,000 cal |
| 5 | 41.84 J |
| 6 | 0.4184 kJ |
| 7 | 2.39 cal |
| 8 | $ 5.02 \times 10^5 $ J |

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