Problem 107 - General Chemistry: Principles and Modern Applications

Question

The reported value for the volume of a rectangular piece of cardboard with the dimensions \(36 \mathrm{cm} \times\) \(20.2 \mathrm{cm} \times 9 \mathrm{mm}\) should be \((\mathrm{a}) \quad 6.5 \times 10^{3} \mathrm{cm}^{3};\) (b) \(7 \times 10^{2} \mathrm{cm}^{3} ;\) (c) \(655 \mathrm{cm}^{3} ;\) (d) \(6.5 \times 10^{2} \mathrm{cm}^{3}\).

Step-by-Step Solution

Answer

The closest reported volume of the rectangular piece of cardboard is \(655 cm^3\).

Step 1: Convert Units

The first crucial step in this exercise is to convert the unit of measurement for thickness from millimeters to centimeters since the other dimensions are already provided in centimeters. This conversion will ensure that all dimensions are in the same unit which is necessary for accurate calculations. To convert millimeters to centimeters, divide by 10. This implies that 9 mm equals 0.9 cm.

Step 2: Apply the formula for volume

After ensuring all measurements are in the same units, proceed to calculate the volume. The volume (V) of a rectangular object is calculated using the formula \(V = length \times width \times height\). Substitute the provided measurements into this formula: \(V = 36 cm \times 20.2 cm \times 0.9 cm\).

Step 3: Calculate the volume

On performing the multiplication \(36 cm \times 20.2 cm \times 0.9 cm\), you get a volume of 653.52 cubic centimeters. However, this result doesn't match exactly to any of the provided options. Hence, the closest, and thereby, the correct answer should be 655 cubic centimeters.

Key Concepts

Unit Conversion

When dealing with volume calculations, having consistent measurement units is key. In this exercise, dimensions are given in both centimeters and millimeters. Converting all measurements to a single unit avoids inaccuracies in computation.
To convert millimeters (mm) to centimeters (cm), use the simple conversion factor:

• 1 cm = 10 mm
• 9 mm = 9 / 10 cm = 0.9 cm

By converting 9 mm to 0.9 cm, all dimensions are now in centimeters, making the calculations straightforward. Remember, consistency in units results in more accurate and reliable results.

Rectangular Prism

A rectangular prism is a 3D shape with six faces, all of which are rectangles. Understanding this shape is fundamental in calculating volume. The volume formula for a rectangular prism is:
\[ V = \text{length} \times \text{width} \times \text{height} \]
In our exercise, these dimensions are:

• Length = 36 cm
• Width = 20.2 cm
• Height = 0.9 cm

By plugging these numbers into the formula, we can calculate the prism's volume systematically. Recognizing the shape is as important as knowing its mathematical representation.

Measurement Accuracy

Achieving accurate calculations requires precise measurement inputs. For this exercise, the calculated volume was 653.52 cm³. Yet, exact answers in mathematical problems sometimes do not match given options perfectly.
Several factors can affect measurement accuracy:

• Rounding errors during calculations
• Significant figures used
• Measurement tools precision

When the calculated result is close to provided options, selecting the nearest option is standard practice. Here, selecting 655 cm³ aligns with realistic mathematical rounding and conventions for significant figures.