Question

To ship figurines, the figurines are placed in a rectangular box and then small packing pellets are added. The base of the box has an area of \(4.4\) in. \(^2\), and the height of the box is \(6.5 \mathrm{in}\). If the volume of the air in the box after the figures and pellets are added is \(8.0 \mathrm{in}^3\), which of the following is closest to the combined volume of the figurines and pellets in the box? A) \(1.9\) in. \(^3\) B) \(20.6\) in \(^3\) C) \(28.6\) in \(^3\) D) \(117.84 \mathrm{in.}^3\)

Short answer

The combined volume of the figurines and pellets in the box is approximately \(20.6 in^3\).

Step-by-step solution

Calculate the total volume of the box

Using the formula for the volume of a rectangular prism, we have: Volume = Base Area × Height. The base area of the box is \(4.4\) in.\(^2\), and the height of the box is \(6.5\) in. So, the total volume of the box is: \(V = 4.4 in^2 \times 6.5 in = 28.6 in^3\)

Calculate the volume of figurines and pellets

To find the combined volume of the figurines and pellets, we need to subtract the volume of the air from the total volume of the box: \(Combined\ volume = Total\ volume - Air\ volume = 28.6 in^3 - 8.0 in^3\)

Calculate the difference

Subtract 8.0 in\(^3\) from the total volume of 28.6 in\(^3\): \(28.6 in^3 - 8.0 in^3 = 20.6 in^3\)

Choose the closest answer

Now, we compare the combined volume we calculated to the given options. Option B is \(20.6 in^3\), which is the closest to our calculated volume of figurines and pellets. Therefore, option B is the correct answer.

Key concepts

Volume Calculation

Volume calculation involves finding out the amount of space that an object occupies. When you are dealing with three-dimensional shapes, the volume tells you how much space is inside the object. For a simple shape like a rectangular prism, the volume can be easily calculated.
The formula to find the volume of a rectangular prism is given by:

• Multiply the area of the base by the height.
  This can be expressed more formally as: \[\text{Volume} = \text{Base Area} \times \text{Height}\]

For example, if the base area of a box is 4.4 square inches and the height is 6.5 inches, the volume of that box is 28.6 cubic inches. It's important to use the proper units (like cubic inches) since volume is a measure of three-dimensional space.

Rectangular Prism

The rectangular prism is one of the most common three-dimensional shapes. It is sometimes referred to as a box shape.
A rectangular prism has the following properties:

• Six rectangular faces.
• All opposite faces are equal.
• It has length, width (or breadth), and height dimensions.

To find the volume of a rectangular prism, you rely on its width, length, and height. In practical terms, if you are dealing with a box or a similar object, think about the steps to find the volume:

• Measure or calculate the area of one of its bases. This is typically length multiplied by width.
• Determine the height of the rectangular prism.
• Multiply the base area by the height to get the total volume.

Problem Solving

Problem-solving in mathematics often entails dissecting a problem into smaller, clearer steps.
When you work through a geometry problem, systematic problem-solving can help:

• Identify what you know and what you need to find. In this case, you have the total volume and need to find the additional space occupied by figurines and pellets.
• Use known formulas or equations to express relationships. Here, you know that the volume taken by air plus figurines and pellets equals the total box volume.
• Substitute the known values into your equations. For example, using the total volume (28.6 cubic inches) and subtracting the air volume (8 cubic inches) gives you the combined volume of 20.6 cubic inches specifically for the figurines and packing pellets.

Breaking down problems this way ensures that you approach them methodically. It also helps to double-check steps and calculations to ensure accuracy.

Geometry

Geometry is an area of mathematics that deals with shapes, sizes, and the properties of space.
In the context of volume calculation, geometry allows you to understand and measure three-dimensional objects like spheres, cylinders, and rectangular prisms. Some key geometric concepts include:

• Surface area vs. volume: Surface area measures the total area of a shape's surface, while volume measures the space inside it.
• Formulas: Each geometric shape has specific formulas for calculating volume. For a rectangular prism, the volume formula is straightforward: base area multiplied by height.
• Applications: Understanding geometry is key for practical tasks such as packaging, construction, or designing objects to fit within space limitations.

Geometry's principles are foundational for solving complex problems and creating structures or models in both academic scenarios and real-world applications.