If the volume of a tank is 4{,}750 text{ cubic centimeters}, what is the volume of the tank in cubic...

1. Translate the problem requirements: We need to convert a volume from cubic centimeters to cubic meters using the given linear conversion factor (\(1 \mathrm{cm} = 0.01 \mathrm{m}\))
2. Understand the cubic conversion relationship: Recognize that volume conversions require cubing the linear conversion factor since volume is three-dimensional
3. Calculate the volume conversion factor: Determine how many cubic centimeters equal one cubic meter
4. Apply the conversion to get the final answer: Divide the given volume by the conversion factor to express it in cubic meters

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find.

We have:

• Tank volume: \(4,750 \mathrm{cm}^3\)
• Conversion fact: \(1 \mathrm{cm} = 0.01 \mathrm{m}\)

We need to find: The tank's volume in cubic meters

Think of it this way - we're taking a measurement that's in small units (cubic centimeters) and expressing it in much larger units (cubic meters). Since we're moving to larger units, we expect our number to get much smaller.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Understand the cubic conversion relationship

Here's the key insight that many students miss: when we convert volume (which involves three dimensions), we can't just use the linear conversion factor directly.

Let's think about this with a concrete example. Imagine a small cube that's 1 centimeter on each side:

• In centimeters: \(1 \mathrm{cm} \times 1 \mathrm{cm} \times 1 \mathrm{cm} = 1 \mathrm{cm}^3\)
• In meters: \(0.01 \mathrm{m} \times 0.01 \mathrm{m} \times 0.01 \mathrm{m} = ? \mathrm{m}^3\)

Notice how we need to convert each dimension separately. Since volume involves length × width × height, and each of these dimensions gets converted using the factor 0.01, we end up multiplying 0.01 three times.

This means: \(1 \mathrm{cm}^3 = 0.01 \times 0.01 \times 0.01 \mathrm{m}^3\)

3. Calculate the volume conversion factor

Now let's work out exactly how many cubic centimeters fit into one cubic meter.

From our understanding above:
\(1 \mathrm{cm}^3 = 0.01 \times 0.01 \times 0.01 \mathrm{m}^3\)
\(1 \mathrm{cm}^3 = 0.000001 \mathrm{m}^3\)

Flipping this relationship around:
\(1 \mathrm{m}^3 = 1,000,000 \mathrm{cm}^3\)

This makes intuitive sense - a cubic meter is much, much larger than a cubic centimeter, so it takes a million cubic centimeters to fill one cubic meter.

4. Apply the conversion to get the final answer

Now we can convert our tank's volume from cubic centimeters to cubic meters.

We use the relationship: \(\mathrm{Volume_{m^3}} = \mathrm{Volume_{cm^3}} \div 1,000,000\)

\(\mathrm{Volume_{m^3}} = 4,750 \div 1,000,000\)
\(\mathrm{Volume_{m^3}} = 0.00475\)

Let's verify this makes sense: we started with 4,750 (a number in the thousands) and ended up with 0.00475 (a very small decimal). This confirms we correctly moved from small units to large units.

4. Final Answer

The volume of the tank is 0.00475 cubic meters.

Looking at our answer choices, this matches option A. 0.00475, which confirms our calculation is correct.

Common Faltering Points

Errors while devising the approach

1. Using linear conversion factor directly for volume

Many students see "\(1 \mathrm{cm} = 0.01 \mathrm{m}\)" and immediately think they can convert cubic centimeters to cubic meters by simply multiplying by 0.01. They fail to recognize that volume is a three-dimensional measurement requiring the conversion factor to be cubed (\(0.01^3 = 0.000001\)).

2. Misunderstanding the direction of conversion

Students often get confused about whether to multiply or divide when converting units. Since we're converting from smaller units (cm³) to larger units (m³), the numerical value should become much smaller, but some students expect it to get larger.

Errors while executing the approach

1. Incorrectly calculating the cubic conversion factor

Even when students understand they need to cube the linear conversion factor, they might make arithmetic errors: calculating \(0.01 \times 0.01 \times 0.01\) incorrectly, or miscounting the number of zeros in the result (getting 0.00001 instead of 0.000001).

2. Using wrong division operation

Students may set up the conversion incorrectly, multiplying 4,750 by 0.000001 instead of dividing by 1,000,000, or they might divide by 1,000 instead of 1,000,000, leading to answers like choice C (0.475).

Errors while selecting the answer

1. Choosing answer based on magnitude expectation rather than calculation

After getting 0.00475, some students second-guess themselves because the number seems "too small" and might select choice B (0.0475) or C (0.475), thinking their decimal placement must be wrong without double-checking their conversion factor.