Question

A rectangular block has dimensions \(2.9 \mathrm{~cm} \times 3.5 \mathrm{~cm} \times\) \(10.0 \mathrm{~cm} .\) The mass of the block is \(615.0 \mathrm{~g}\). What are the volume and density of the block?

Short answer

The volume of the block is \(101.5 cm^3\) and its density is approximately \(6.06 g/cm^3\).

Step-by-step solution

Calculate the volume of the block

To find the volume, we will multiply the length, width, and height of the block: Volume = Length × Width × Height Using the given dimensions: Volume = \(2.9 cm × 3.5 cm × 10.0 cm\)

Compute the volume

Now, let's compute the volume by multiplying the dimensions: Volume = \(2.9 cm × 3.5 cm × 10.0 cm = 101.5 cm^3\) So, the volume of the block is \(101.5 cm^3\).

Calculate the density of the block

Now we can calculate the density using the formula: Density = Mass / Volume Given the mass is 615.0 g, and the volume is 101.5 \(cm^3\), we can find the density as follows: Density = \(615.0 g / 101.5 cm^3\)

Compute the density

Let's compute the density: Density = \(615.0 g / 101.5 cm^3 ≈ 6.06 g/cm^3\) So, the density of the block is approximately \(6.06 g/cm^3\). In conclusion, the volume of the block is \(101.5 cm^3\) and its density is approximately \(6.06 g/cm^3\).

Key concepts

Volume of a Rectangular Block

Understanding how to find the volume of a rectangular block is a fundamental concept in geometry and various applications such as packaging, construction, and manufacturing. The volume represents the amount of space the block occupies and is calculated by multiplying its length, width, and height together.

For a rectangular block with dimensions length (\text{l}), width (\text{w}), and height (\text{h}), the volume (\text{V}) is expressed mathematically as:
\[\begin{equation}Volume (V) = length (l) \times width (w) \times height (h)\end{equation}\]
Applying this formula to the given exercise where the block's dimensions are 2.9 cm, 3.5 cm, and 10.0 cm, results in:
\[\begin{equation}Volume = 2.9 \text{cm} \times 3.5 \text{cm} \times 10.0 \text{cm} = 101.5 \text{cm}^3\end{equation}\]
The result, 101.5 cubic centimeters, tells us how much space is enclosed within this block. This type of calculation is also critical when packing objects to optimize space use or when determining the amount of material required to create a solid object.

Mass to Volume Ratio

The mass to volume ratio, also known as density, is a crucial concept in science, particularly in chemistry and physics. It is a measure of how much mass is contained in a given volume and is essential for identifying substances and understanding their properties.

Mathematically, density (\text{D}) is defined as mass (\text{m}) divided by volume (\text{V}):
\[\begin{equation}Density (D) = \frac{Mass (m)}{Volume (V)}\end{equation}\]
This formula allows us to calculate the density of a substance when we have measurements for both its mass and its volume. In the case of the rectangular block in our exercise, with a mass of 615.0 grams and a volume of 101.5 cubic centimeters, the calculation would be:
\[\begin{equation}Density = \frac{615.0 \text{g}}{101.5 \text{cm}^3} \approx 6.06 \text{g/cm}^3\end{equation}\]
The density value of approximately 6.06 grams per cubic centimeter provides us with an understanding of how tightly matter is packed within the block, which can help in the identification of the material of the block and in determining its buoyancy in different fluids.

Chemical Measurement

Chemical measurement, which involves quantifying the characteristics of substances, is a vital component of chemical sciences. Measurements such as mass, volume, and density provide critical information for laboratory work, quality control in manufacturing, and compliance with safety regulations.

The practice of chemical measurement requires accuracy and precision, especially when preparing solutions or conducting experiments. Balances are used to measure mass, while graduated cylinders or volumetric flasks are used to measure volumes accurately.

In our exercise, the rectangular block's mass is precisely given as 615.0 grams. This precise measurement is necessary to calculate the density accurately. High-quality scales capable of measuring to the nearest tenth of a gram are typically used for such tasks. Similarly, volumes must be measured with care, using the correct equipment for the level of precision required.

Understanding these chemical measurement principles not only aids in solving textbook exercises but also forms a solid foundation for future laboratory work and scientific exploration.