Question

a conversion table showing how to convert between \\(\text{mm}^3\\), \\(\text{cm}^3\\) and \\(\text{m}^3\\) is shown.
for example, the value of a will show the conversion of \\(1\text{ mm}^3\\) into \\(\text{cm}^3\\)

what values replace a-f?

describe any patterns you see between the rows and columns in your table.

Explanation:

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

Step 1: Understand linear conversions

To convert between cubic units, we first look at the linear relationships between millimeters (\(\text{mm}\)), centimeters (\(\text{cm}\)), and meters (\(\text{m}\)):

• \(1\text{ cm} = 10\text{ mm}\)
• \(1\text{ m} = 100\text{ cm} = 1,000\text{ mm}\)

Step 2: Convert cubic units by cubing the linear factors

Since volume is three-dimensional, we cube the linear conversion factors:

• From \(\text{cm}^3\) to \(\text{mm}^3\):

$$ (1\text{ cm})^3 = (10\text{ mm})^3 \implies 1\text{ cm}^3 = 1,000\text{ mm}^3 $$

This gives us \(C = 1,000\) (or \(10^3\)).

• From \(\text{mm}^3\) to \(\text{cm}^3\):

$$ 1\text{ mm}^3 = \frac{1}{1,000}\text{ cm}^3 = 0.001\text{ cm}^3 $$

This gives us \(A = 0.001\) (or \(10^{-3}\)).

• From \(\text{m}^3\) to \(\text{cm}^3\):

$$ (1\text{ m})^3 = (100\text{ cm})^3 \implies 1\text{ m}^3 = 1,000,000\text{ cm}^3 $$

This gives us \(F = 1,000,000\) (or \(10^6\)).

• From \(\text{cm}^3\) to \(\text{m}^3\):

$$ 1\text{ cm}^3 = \frac{1}{1,000,000}\text{ m}^3 = 0.000001\text{ m}^3 $$

This gives us \(D = 0.000001\) (or \(10^{-6}\)).

• From \(\text{m}^3\) to \(\text{mm}^3\):

$$ (1\text{ m})^3 = (1,000\text{ mm})^3 \implies 1\text{ m}^3 = 1,000,000,000\text{ mm}^3 $$

This gives us \(E = 1,000,000,000\) (or \(10^9\)).

• From \(\text{mm}^3\) to \(\text{m}^3\):

$$ 1\text{ mm}^3 = \frac{1}{1,000,000,000}\text{ m}^3 = 0.000000001\text{ m}^3 $$

This gives us \(B = 0.000000001\) (or \(10^{-9}\)).

Step 3: Identify patterns in the table

• Symmetry of Reciprocals: The values diagonally opposite across the main diagonal (the \(1\)s) are reciprocals of each other. For example, \(C = 1,000\) and \(A = \frac{1}{1,000}\).
• Powers of 10: Every value in the table is a power of \(10\) (specifically, powers of \(10^3\) or \(1,000\)).
• Scale Factor: Moving one column to the right divides the value by \(1,000\) (or \(1,000,000\) when moving from \(\text{cm}^3\) to \(\text{m}^3\)), and moving one row down multiplies the value by those same factors.

Answer:

Values:

• A = \(0.001\) (or \(10^{-3}\))
• B = \(0.000000001\) (or \(10^{-9}\))
• C = \(1,000\) (or \(10^3\))
• D = \(0.000001\) (or \(10^{-6}\))
• E = \(1,000,000,000\) (or \(10^9\))
• F = \(1,000,000\) (or \(10^6\))

Patterns:

1. Reciprocal Pairs: Values reflected across the main diagonal of \(1\)s are reciprocals of each other (e.g., \(C = 10^3\) and \(A = 10^{-3}\)).
2. Powers of 10: All conversion factors are powers of \(1,000\) (which is \(10^3\)).