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 rectangular block is \(101.5 cm^3\), and its density is \(6.06 \frac{g}{cm^3}\).

Step-by-step solution

Find the volume of the block

To find the volume of the block, we can use the formula for the volume of a rectangular prism, which is: \( Volume = Length × Width × Height\) We are given the dimensions of the block as 2.9 cm × 3.5 cm × 10.0 cm. By substituting these values into the formula, we can calculate the volume: \( Volume = 2.9 cm × 3.5 cm × 10.0 cm\)

Calculate the volume

Now, we can calculate the volume by multiplying all the given dimensions: \( Volume = 2.9 cm × 3.5 cm × 10.0 cm = 101.5 cm^3\) So, the block has a volume of 101.5 cm³.

Find the density of the block

As we know, the formula to find the density is: \( Density = \frac{Mass}{Volume}\) We are given the mass of the block as 615.0 g, and we just calculated the volume to be 101.5 cm³. By substituting these values into the formula, we can calculate the density: \( Density = \frac{615.0 g}{101.5 cm^3}\)

Calculate the density

Now, we can calculate the density of the block by dividing the mass by the volume: \( Density = \frac{615.0 g}{101.5 cm^3} = 6.06 \frac{g}{cm^3}\) So, the block has a density of 6.06 g/cm³. In conclusion, the volume of the block is 101.5 cm³, and its density is 6.06 g/cm³.

Key concepts

Volume Calculation

When you have a block and you need to figure out how much space it takes up, you're talking about its volume. This can be found using the volume formula for a rectangular prism. For a block with three lines of measurements, you multiply them altogether. So, if you know the block is 2.9 cm wide, 3.5 cm tall, and 10.0 cm long, the volume is simply the product of these three measurements. The formula you use is: \[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \] Plugging in the numbers, \[ 2.9 \text{ cm} \times 3.5 \text{ cm} \times 10.0 \text{ cm} = 101.5 \text{ cm}^3 \] This means the block occupies 101.5 cubic centimeters of space.

Rectangular Prism

A rectangular prism is a 3D shape much like a box. It has six sides, all in the form of rectangles. Each of these rectangles is defined by the prism's length, width, and height. Think of it as a stretched out rectangle, expanded into three-dimensional space. Key features include:

• Opposite faces are equal and parallel.
• It can also be known as a cuboid.
• All angles are right angles (90 degrees).

Knowing these properties helps with calculating its volume, as it's just about multiplying the three dimensions that define its size.

Mass and Volume Relationship

The connection between mass and volume is crucial for understanding material properties. Mass is how heavy something is, while volume is how much space it occupies. The relationship between the two helps us understand how particles inside an object are packed together. For example, a heavy object with a small volume is quite dense compared to a lighter object with the same volume. This relationship is key when calculating density, as we need both mass (given as 615.0 g) and volume (calculated as 101.5 cm³) to know how tightly packed the object is.

Density Formula

Density is like a comparison of how much mass fits into a certain volume. It tells us how closely packed the material is in an object. The formula to find density combines what you learned about mass and volume. It is: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] So, if you know the block's mass is 615.0 grams and it occupies a volume of 101.5 cm³, the density calculation is straightforward: \[ \text{Density} = \frac{615.0 \text{ g}}{101.5 \text{ cm}^3} = 6.06 \frac{\text{g}}{\text{cm}^3} \] This means each cubic centimeter of the block weighs 6.06 grams, helping you understand the material's compactness.