Problem 65 - Algebra 2

Question

REVIEW How many 5 -inch cubes can be placed completely inside a box that is 10 inches long, 15 inches wide, and 5 inches tall? $$ \begin{array}{ll}{\text { F } 5} & {\text { H } 20} \\ {\text { G } 6} & {\text { J } 15}\end{array} $$

Step-by-Step Solution

Answer

The box can hold 6 cubes.

Step 1: Calculate the Volume of the Box

The box has dimensions 10 inches by 15 inches by 5 inches. Multiply the length, width, and height to find the volume of the box: \(10 \times 15 \times 5 = 750 \text{ cubic inches}\).

Step 2: Calculate the Volume of One Cube

The cube has a side length of 5 inches. Calculate the volume of the cube by cubing the side length: \(5 \times 5 \times 5 = 125 \text{ cubic inches}\).

Step 3: Determine Maximum Number of Cubes

To find out how many 5-inch cubes can fit inside the box, divide the volume of the box by the volume of one cube: \(\frac{750}{125} = 6\).

Step 4: Verify Fit Based on Box Dimensions

Check to ensure the cubes can fit based on arrangement: The box is 10 inches long, 15 inches wide, and 5 inches high. Two cubes fit along the 10-inch length, 3 cubes along the 15-inch width, and exactly 1 cube fits in the 5-inch height. This arrangement confirms 6 cubes can fit based on dimensions.

Key Concepts

Geometry

Geometry is a branch of mathematics focusing on the properties and relationships of points, lines, surfaces, and solids. When dealing with problems in geometry, you often work with a variety of shapes and try to calculate their properties, such as area and volume.

The shapes can range from simple figures like squares and rectangles to more complex forms like prisms and spheres. In geometry, understanding the dimensions of a shape (like its length, width, and height) is crucial for solving problems. This understanding enables the calculation of other important properties that describe the shape.

Geometry not only involves the study of each shape but also provides methods to transform one geometric shape into another. By using basic geometric principles, students can learn how to manage space efficiently, solve real-world architectural problems, and comprehend the spatial relationships between different objects.

Volume of Rectangular Prism

A rectangular prism is a three-dimensional solid object which has six faces, all of which are rectangles. To find the volume of a rectangular prism, you need to multiply its length, width, and height together. This mathematical formula is expressed as follows:

\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]

For example, when you have a box measuring 10 inches long, 15 inches wide, and 5 inches high, you calculate the volume by multiplying these dimensions: \(10 \times 15 \times 5 = 750\) cubic inches.

Understanding the concept of volume in geometry helps you determine the capacity of an object or how much space it occupies, which is fundamental when tackling problems related to space management in daily life, like fitting objects into a container. This particularly applies to determining how many smaller objects can fit inside a larger container, as shown in the given problem.

Volume of Cube

A cube is a special type of rectangular prism where all sides are of equal length. Calculating its volume is straightforward because you only need to know the length of one side. The formula to find a cube’s volume is:

\[ \text{Volume} = \text{side length}^3 \]

Consider a cube with each side measuring 5 inches. The volume would be calculated as: \(5 \times 5 \times 5 = 125\) cubic inches. This means that the cube occupies 125 cubic inches of space.

Understanding the volume of cubes is essential for many spatial reasoning problems, such as the one where you determine how many such cubes can fit inside a particular space. When the dimensions of a cube fit snugly into the dimensions of another geometric shape, like a box, it simplifies calculations because each cube can be envisaged as filling up perfectly defined units of space within the larger shape's volume.