If the volume of a small container is 14{,}520text{ cubic millimeters}, what is the volume of the container in cubic...

1. Translate the problem requirements: We need to convert 14,520 cubic millimeters to cubic centimeters using the given conversion rate of \(1 \text{ millimeter} = 0.1 \text{ centimeter}\)
2. Establish the volume conversion relationship: Determine how the linear conversion factor (0.1) applies to volume conversion by understanding that volume involves three dimensions
3. Apply the conversion factor: Use the cubed conversion factor to transform the given volume from cubic millimeters to cubic centimeters
4. Verify against answer choices: Confirm our result matches one of the provided options

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're being asked to do. We have a container with a volume of 14,520 cubic millimeters, and we need to find out what that same volume is when measured in cubic centimeters.

We're given the conversion rate: \(1 \text{ millimeter} = 0.1 \text{ centimeter}\). This means that every millimeter is one-tenth of a centimeter.

The key insight here is understanding what "cubic" means. When we talk about cubic millimeters or cubic centimeters, we're talking about volume - the amount of space something takes up in three dimensions: length, width, and height.

2. Establish the volume conversion relationship

Here's where we need to think carefully. When we convert from millimeters to centimeters for length, we multiply by 0.1. But volume is three-dimensional!

Let's think about this with a simple example: Imagine a cube that's 1 millimeter on each side. Its volume would be \(1 \times 1 \times 1 = 1\) cubic millimeter.

Now, if we convert each side to centimeters: 1 mm = 0.1 cm. So our cube becomes 0.1 cm × 0.1 cm × 0.1 cm in size.

The volume of this same cube in cubic centimeters would be: \(0.1 \times 0.1 \times 0.1 = 0.001\) cubic centimeters.

Notice that we had to use the conversion factor three times (once for each dimension), which means we cube the linear conversion factor: \((0.1)^3 = 0.001\)

So to convert from cubic millimeters to cubic centimeters, we multiply by \((0.1)^3 = 0.001\)

3. Apply the conversion factor

Now we can apply our conversion factor to the given volume:

Volume in cubic centimeters = Volume in cubic millimeters × 0.001

Volume in cubic centimeters = \(14,520 \times 0.001\)

When we multiply by 0.001, we're essentially moving the decimal point three places to the left:

\(14,520 \times 0.001 = 14.520\)

Therefore, the volume is 14.52 cubic centimeters.

4. Verify against answer choices

Looking at our answer choices:

1. 0.1452
2. 1.452
3. 14.52
4. 145.2
5. 1,452

Our calculated answer of 14.52 cubic centimeters matches choice C exactly.

We can double-check this makes sense: since centimeters are larger units than millimeters, we expect the numerical value to be smaller when we convert to cubic centimeters, which it is (\(14.52 < 14,520\) ✓).

Final Answer

The volume of the container is 14.52 cubic centimeters.

Answer: C

Common Faltering Points

Errors while devising the approach

1. Using linear conversion factor instead of cubic conversion factor: Students often fail to recognize that volume is three-dimensional and incorrectly apply the linear conversion factor (0.1) directly instead of cubing it. They might think: "1 mm = 0.1 cm, so 14,520 cubic mm = \(14,520 \times 0.1 = 1,452\) cubic cm" leading to answer choice E.

2. Confusing the conversion direction: Students may incorrectly think that since centimeters are larger units, they need to multiply by a factor greater than 1 to convert from the smaller unit (mm³) to the larger unit (cm³), not realizing that the numerical value should decrease when converting to larger units.

3. Misunderstanding the relationship between linear and volume measurements: Students might not grasp that when converting volume (cubic units), they need to apply the linear conversion factor three times (once for each dimension: length × width × height), leading them to use incorrect conversion approaches.

Errors while executing the approach

1. Arithmetic errors with decimal multiplication: Even if students correctly identify that they need to multiply by 0.001, they may make computational errors when calculating \(14,520 \times 0.001\), potentially misplacing the decimal point or making multiplication mistakes.

2. Incorrectly calculating \((0.1)^3\): Students might miscalculate the cubic conversion factor, computing \((0.1)^3\) as 0.01 instead of 0.001, which would lead them to get \(14,520 \times 0.01 = 145.2\) (answer choice D).

Errors while selecting the answer

No likely faltering points - the calculated result directly matches one of the answer choices, and there's no additional interpretation or rounding required for the final answer selection.