A box measures 3.12 ft in length, 0.0455 yd in width, and 7.87 inches in height.  What is its volume in mm^3 if the volume with correct sig figs?

To calculate the volume of the box, we first need to convert all measurements to the same unit (millimeters, mm) and then compute the volume using the formula for the volume of a rectangular box: 

\[
V = \text{length} \times \text{width} \times \text{height}
\]

### Step 1: Convert all measurements to millimeters (mm)

- **Length**: 3.12 feet to millimeters  
  \[
  1 \text{ ft} = 304.8 \text{ mm} \implies 3.12 \text{ ft} = 3.12 \times 304.8 = 950.976 \text{ mm}
  \]

- **Width**: 0.0455 yards to millimeters  
  \[
  1 \text{ yd} = 914.4 \text{ mm} \implies 0.0455 \text{ yd} = 0.0455 \times 914.4 = 41.5752 \text{ mm}
  \]

- **Height**: 7.87 inches to millimeters  
  \[
  1 \text{ inch} = 25.4 \text{ mm} \implies 7.87 \text{ in} = 7.87 \times 25.4 = 199.898 \text{ mm}
  \]

### Step 2: Calculate the volume in cubic millimeters

\[
V = 950.976 \text{ mm} \times 41.5752 \text{ mm} \times 199.898 \text{ mm}
\]

Now, let's compute the volume and round to the correct number of significant figures. The least number of significant figures in the given measurements is 3 (from the width: 0.0455).

### Step 3: Perform the calculation

Let's calculate the volume:  
\[
V \approx 950.976 \times 41.5752 \times 199.898
\]The converted dimensions in millimeters are:

- **Length**: \(950.976 \, \text{mm}\)
- **Width**: \(41.6052 \, \text{mm}\)
- **Height**: \(199.898 \, \text{mm}\)

### Volume Calculation

\[
V = 950.976 \times 41.6052 \times 199.898 \approx 7,909,073.649 \, \text{mm}^3
\]

### Step 4: Round to the Correct Significant Figures

The measurement with the fewest significant figures is 0.0455 (3 significant figures). Thus, the volume should be rounded to 3 significant figures:

\[
V \approx 7.91 \times 10^6 \, \text{mm}^3
\]

### Final Answer

The volume of the box, with the correct number of significant figures, is **\(7.91 \times 10^6 \, \text{mm}^3\)**.

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How do you convert between different units of volume?
2. What are significant figures, and why are they important in calculations?
3. How would you calculate the surface area of this box?
4. How does dimensional analysis help in unit conversions?
5. What other physical properties could be calculated for this box using its dimensions?

**Tip:** Always check the number of significant figures when converting units to ensure precision and accuracy in scientific calculations.

Learn how to calculate the volume of a box measuring 3.12 ft in length, 0.0455 yd in width, and 7.87 inches in height in cubic millimeters with correct significant figures. Understand the unit conversion process to millimeters and apply the volume formula. Suitable for high school students studying unit conversion and precision in measurements.

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