Question

A piece of green jade was cut into a rectangular solid measuring \(2.50 \mathrm{~cm}\) by \(1.25 \mathrm{~cm}\) by \(3.50 \mathrm{~mm}\). What is the volume in cubic centimeters?

Short answer

The volume of the rectangular solid is 1.09 cm³.

Step-by-step solution

Convert all measurements to the same unit

The height of the rectangular solid is given in millimeters, so we need to convert it to centimeters. Since there are 10 millimeters in a centimeter, 3.50 mm is equivalent to \( \frac{3.50}{10} = 0.35 \) cm.

Understand the formula for volume

The volume \( V \) of a rectangular solid is calculated as the product of its length \( l \), width \( w \), and height \( h \). The formula is: \[ V = l \times w \times h \]

Substitute values into the volume formula

Insert the dimensions of the rectangular solid into the volume formula: \( V = 2.50 \text{ cm} \times 1.25 \text{ cm} \times 0.35 \text{ cm} \).

Perform the calculation

Multiply the given dimensions:\[ V = 2.50 \times 1.25 \times 0.35 \]First, calculate \( 2.50 \times 1.25 = 3.125 \). Then multiply the result by 0.35: \( 3.125 \times 0.35 = 1.09375 \).

State the answer in correct units

The volume of the rectangular solid is \( 1.09375 \) cubic centimeters. Since volumes are typically rounded based on significant figures, you may choose to report this to the correct number of significant figures based on the given data (which is three significant figures): \( 1.09 \) cm³.

Key concepts

Unit Conversion

Unit conversion is a process of changing measurements from one unit to another. In our exercise, we deal with two units: centimeters (cm) and millimeters (mm). Understanding this concept is essential in science and mathematics to ensure consistency.

• To convert from millimeters to centimeters, you'll divide the number of millimeters by 10, because there are 10 millimeters in a centimeter.
• For example, converting 3.50 mm into centimeters involves the calculation: \( \frac{3.50}{10} = 0.35 \) cm.

This ensures measurements are in the same units before performing operations like volume calculations, which require consistency to yield accurate results.

Significant Figures

Significant figures represent the digits in a measurement that carry meaning about its precision. They are crucial in scientific calculations to indicate accuracy.

• For example, the lengths 2.50 cm and 1.25 cm both have three significant figures, indicating a precision to one-hundredth of a centimeter.
• In our example, the calculated volume is 1.09375 cm³, but based on the original measurements, we can only confidently report this value to three significant figures from the initial measurements.
• Thus, the volume is simplified to 1.09 cm³ to reflect this accuracy.

This maintains consistency and accuracy in calculated results.

Rectangular Solid Volume

Finding the volume of a rectangular solid involves multiplying its length, width, and height. This formula is straightforward:\[ V = l \times w \times h \] where \( V \) is volume, \( l \) is the length, \( w \) is the width, and \( h \) is the height.
Applying this to our rectangular solid:

• We multiply the measurements: \( V = 2.50 \times 1.25 \times 0.35 \).
• First, calculate \( 2.50 \times 1.25 = 3.125 \), then \( 3.125 \times 0.35 = 1.09375 \) cm³.

This formula provides a way to easily calculate the space occupied by a rectangular object.

Measurement Conversion

Measurement conversion ties closely to unit conversion, but it also involves converting between different types of measures like dimensions to volume.

• First, convert all dimensions to the same units to ensure they can be multiplied together correctly.
• For instance, converting a height given in millimeters to centimeters aligns it with length and width measurements in centimeters.
• Understanding these conversions facilitates accurate volume calculations and avoids potential errors in scientific work.

Ensuring such consistency is pivotal for accurate real-world applications and calculations.

Cubic Centimeters

The volume of solids is commonly expressed in cubic centimeters (cm³), a unit representing a cube with 1 cm sides.

• This unit is particularly useful in measuring volumes in labs and science, where the metric system is predominantly used.
• A cubic centimeter corresponds to exactly one milliliter (mL) in fluid measures, which is another metric advantage for liquid-solid conversions.
• In our problem, the final volume of the jade stone was calculated as 1.09375 cm³.

By using cubic centimeters, we align with standard scientific conventions, simplifying communication and comparison of measurements.