Book cover for McDougal Littell Jurgensen Geometry : Student Edition 2000

McDougal Littell Jurgensen Geometry : Student Edition 2000

Ray C. Jurgensen, Richard G. Brown

ISBN #9780395977279

1st Edition

2,712 Questions

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20,674 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers fundamental concepts for calculating areas and volumes of various solids such as prisms, pyramids, cylinders, cones, and spheres. It explains the importance of understanding similar solids and scaling relationships for accurate volume calculations. Additionally, pivotal principles like Cavalieri's Principle are introduced, and applications in technology and real-world design, including geodesic domes and the work of Buckminster Fuller, provide context and practical relevance to the mathematical formulas and procedures discussed.

Learning Objectives

1

Explain and apply the formulas for calculating areas and volumes of important solids including prisms, pyramids, cylinders, cones, and spheres.

2

Analyze the properties of similar solids and understand how scaling affects their areas and volumes.

3

Demonstrate the application of fundamental principles such as Cavalieri's Principle in determining volumes.

4

Utilize technology and computer tools for solving geometric problems related to areas and volumes.

5

Integrate real-world applications, such as in geodesic dome design and Buckminster Fuller's work, to contextualize mathematical concepts.

Key Concepts

CONCEPT

DEFINITION

Prism

A solid figure with two parallel, congruent faces (bases) and rectangular side faces. Its volume is calculated by multiplying the area of the base by the height.

Pyramid

A solid figure with a polygonal base and triangular faces that meet at a common point called the apex. Its volume is one-third of the product of the base area and the height.

Cylinder

A solid with two parallel circular bases connected by a curved surface. Its volume is determined by the area of the base multiplied by the height.

Cone

A solid with a circular base that tapers smoothly to a point called the apex. Its volume is one-third of the product of the base area and the height.

Sphere

A perfectly round solid where every point on the surface is equidistant from the center. Its volume is calculated using the formula (4/3)πr³.

Similar Solids

Solids that have the same shape but different sizes. Their corresponding linear dimensions are proportional, and areas and volumes are scaled by the square and cube of the scale factor, respectively.

Cavalieri's Principle

A principle stating that if two solids have the same height and cross-sectional area at every level, they have the same volume.

Geodesic Dome

A spherical or partial-spherical structure composed of a network of triangles, popularized by Buckminster Fuller, noted for efficiency and strength.

Example Problems

Example 1

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table. (TABLE CAN'T COPY).

Example 2

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table. (TABLE CAN'T COPY).

Example 3

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table. (TABLE CAN'T COPY).

Example 4

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table. (TABLE CAN'T COPY).

Example 5

Exercises $1-6$ refer to rectangular solids with dimensions $l, w,$ and $h .$ Complete the table. (TABLE CAN'T COPY).
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Step-by-Step Explanations

QUESTION

How do you calculate the volume of a cylinder?

STEP-BY-STEP ANSWER:

Step 1: Identify the radius (r) of the circular base and the height (h) of the cylinder.
Step 2: Calculate the area of the base using the formula A = πr².
Step 3: Multiply the area of the base by the height of the cylinder, using the formula V = πr²h.
Final Answer: The volume of the cylinder is V = πr²h.

Volume of a Cylinder

QUESTION

How is the volume of a cone calculated?

STEP-BY-STEP ANSWER:

Step 1: Identify the radius (r) of the circular base and the height (h) of the cone.
Step 2: Compute the area of the base using the formula A = πr².
Step 3: Use the formula for the volume of a cone: V = (1/3)πr²h.
Final Answer: The volume of the cone is V = (1/3)πr²h.

Volume of a Cone

QUESTION

How does scaling affect the volumes of similar solids?

STEP-BY-STEP ANSWER:

Step 1: Determine the scale factor (k) between two similar solids by comparing any pair of corresponding linear dimensions.
Step 2: Understand that areas will scale by a factor of k² and volumes by a factor of k³.
Step 3: Apply the scaling relationship to find an unknown volume if the volume of one solid is known.
Final Answer: The volume of the scaled solid is found by multiplying the original volume by k³.

Scaling in Similar Solids

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Common Mistakes

  • Mixing up the formulas for different solids (e.g., confusing the volume formula for a cylinder with that of a cone).
  • Forgetting to apply the proper scaling factor—especially squaring the scale factor for areas and cubing it for volumes when working with similar solids.
  • Omitting the one-third factor in the volume calculations for pyramids and cones.
  • Misapplying Cavalieri's Principle by not ensuring that cross-sectional areas are congruent at every level.