Question

write the following volumes in ascending order:

\\(17\text{ cm}^3\\) \\(0.0021\text{ cm}^3\\) \\(1.2\text{ mm}^3\\) \\(14\\,000\text{ mm}^3\\)

Explanation:

🆕 New Concept Discovered: Converting Cubic Units
Converting volume units requires cubing the linear conversion factor.

Step 1: Identify the conversion factor

To compare the volumes, we need to convert them all to the same unit. Let's convert everything to cubic millimeters (\(\text{mm}^3\)).

We know the linear relationship between centimeters and millimeters:

$$ 1\text{ cm} = 10\text{ mm} $$

Since volume is three-dimensional, we cube this conversion factor to convert cubic units:

$$ 1\text{ cm}^3 = (10\text{ mm})^3 = 1000\text{ mm}^3 $$

This means to convert from \(\text{cm}^3\) to \(\text{mm}^3\), we multiply by \(1000\).

Step 2: Convert all values to cubic millimeters

Now, we convert each value given in \(\text{cm}^3\) into \(\text{mm}^3\):

• First value: \(17\text{ cm}^3\)

$$ 17 \times 1000 = 17\,000\text{ mm}^3 $$

• Second value: \(0.0021\text{ cm}^3\)

$$ 0.0021 \times 1000 = 2.1\text{ mm}^3 $$

• Third value: \(1.2\text{ mm}^3\) (already in \(\text{mm}^3\))

• Fourth value: \(14\,000\text{ mm}^3\) (already in \(\text{mm}^3\))

Step 3: Order the values from smallest to largest

Now we compare the values in \(\text{mm}^3\):

$$ 1.2\text{ mm}^3 < 2.1\text{ mm}^3 < 14\,000\text{ mm}^3 < 17\,000\text{ mm}^3 $$

Replacing them with their original forms:

• \(1.2\text{ mm}^3\)
• \(0.0021\text{ cm}^3\) (which is \(2.1\text{ mm}^3\))
• \(14\,000\text{ mm}^3\)
• \(17\text{ cm}^3\) (which is \(17\,000\text{ mm}^3\))

Answer:

$$ 1.2\text{ mm}^3,\quad 0.0021\text{ cm}^3,\quad 14\,000\text{ mm}^3,\quad 17\text{ cm}^3 $$