Problem 13 Draw a net of each solid shown o... [FREE SOLUTION]

Chapter 11: Problem 13

Draw a net of each solid shown or described. Then find the lateral area and surface area of each solid. Round to the nearest tenth, if necessary. cube: side length \(7 \mathrm{ft}\)

Short Answer

Expert verified

Draw six squares in a cross shape. The lateral area is 196 ft虏, and the surface area is 294 ft虏.

Step by step solution

Identifying the Net of the Cube

A cube has six faces, all of which are squares. The net of a cube consists of these six squares arranged in an open layout such that they can be folded to form the cube. Imagine a cross shape where one square is at the center, and it is surrounded by four squares on its sides, with the sixth square attached to any of the faces.

Calculating the Lateral Area (LA)

The lateral area of a solid refers to the sum of the areas of all the sides except the top and bottom. For a cube, the lateral area is the sum of four side faces. Since each face is a square with side length 7 ft, the area of one face is \(7 \times 7 = 49 \, \text{ft}^2\). Thus, the lateral area is \(4 \times 49 = 196 \, \text{ft}^2\).

Calculating the Surface Area (SA)

The surface area of a cube is the sum of the areas of all six of its faces. Using the same face area calculation as in the lateral area, the surface area of the cube is \(6 \times 49 = 294 \, \text{ft}^2\).

Rounding the Areas

The calculated lateral area is 196 ft虏 and the surface area is 294 ft虏. Since these results are already whole numbers, no rounding is needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Cube

A cube is a three-dimensional geometric shape. It has six faces, with all sides equal in length. Each face is a square. This means all angles in a cube are right angles, and every edge is of the same length. For the cube in this exercise, each edge measures 7 feet. This uniformity makes calculations like lateral and surface areas straightforward, as each face contributes equally to these measurements.

In real life, cubes are found in objects like dice and certain packaging boxes. In geometry, understanding cubes is vital, as they serve as a basis for more complex polyhedra.

Exploring the Net of a Solid

A net is a two-dimensional representation of a three-dimensional figure. For a cube, the net is an arrangement of its six square faces. The net lays flat and visually represents how the cube looks before it's folded into its 3D form.

When constructing or visualizing a net of a cube, consider a cross-shaped pattern. Place one square in the center with another square attached to each of its four edges. The final square can attach to any of the others. This way, the net, when folded properly, constructs a complete cube.

The net helps in understanding the structure of the cube.
It aids in calculating both lateral and surface areas.
Practice drawing the net to sharpen spatial visualization skills.

Demystifying Lateral Area

The lateral area of a solid primarily involves the area of its sides. For a cube, it consists of the four vertical faces that connect the top and bottom. Computing the lateral area involves calculating the area of these four faces.

Given that each face is a square with a side length of 7 feet, its area is simple to find: multiply 7 by 7, resulting in 49 square feet per side. For the lateral area:

Multiply this area by 4, accounting for all sides except the top and bottom.
Lateral area = 4 x 49 = 196 square feet.
This serves as crucial information when only the wrap-around part of the cube is stripped away.

Unpacking Surface Area

The surface area is the measure of the total area of all the surfaces of a three-dimensional shape. In the context of a cube, it means adding up the area of all six faces.

Since the cube has square faces of equal dimensions鈥�7 feet by 7 feet in this instance鈥攜ou calculate one face to be 49 square feet. Thus, for the entire surface area:

Multiply the area of one face (49 sq ft) by 6, since there are six identical faces.
Surface area = 6 x 49 = 294 square feet.
These calculations ensure you account for the whole outer layer of the cube.

Understanding surface area is particularly useful in practical applications, such as determining the amount of material needed to cover a cube, like wrapping a present or painting a box.

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91影视

Draw a net of each solid shown or described. Then find the lateral area and surface area of each solid. Round to the nearest tenth, if necessary. cube: side length \(7 \mathrm{ft}\)

Short Answer

Step by step solution

Identifying the Net of the Cube

Calculating the Lateral Area (LA)

Calculating the Surface Area (SA)

Rounding the Areas

Key Concepts

Understanding the Cube

Exploring the Net of a Solid

Demystifying Lateral Area

Unpacking Surface Area

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