Foundations of Algebra and Geometry
Chapter 5 Answers
Spatial Relations
Artists and architects use mathematics both to design and to create their works.
Part B, Comparing Sizes
1. You can use the Web to look at artists' works and discover how they are able to represent relative size.
a. Study the painting The Annunciation by the Italian Renaissance painter Leonardo da Vinci. Find the horizon line, the vanishing point, and orthogonal lines. Then click on the buttons to see if you were right.
b. Study the painting Little Street by the Dutch painter Jan Vermeer. Explain how to find the vanishing point.
[Answers will vary. Use the chimneys on the left side of the painting to draw diagonals that meet at the vanishing point.]c. If the artist was across the street from the building, on what floor would you say he was? Explain your choice.
[Answers will vary. All diagonal lines appear to meet in a second floor window in the painting. Since the horizon line in a perspective drawing indicates eye level, the artist was probably on the second floor of the building across the street from the building in the painting.]d. Study the drawing of Six cubes. Name ways that, along with perspective, an artist can use to create the illusion of three dimensions.
[Answers will vary. As objects get farther and farther away, show them shrinking in size, growing closer together, and losing detail and color so that they appear "hazier."]e. How has the artist made the Sculpture behind the collonade appear larger than it really is?
[Answers will vary. The photo of the two women shows that the opening at the back of the colonnade is much lower and smaller than that at the front. The colonnade actually "shrinks" front-to-back, creating a fake perspective and the illusion that an object framed in the rear doorway must be much taller than it actually is.]f. Copy the rectangles on a sheet of paper. Then complete the drawing in a way that shows that the rectangle on the left is an object in the background which is larger than the object represented by the rectangle on the right (in the foreground).
[Drawings will vary. One approach is to create two objects, the left one of which is a large object placed farther from the eye than a smaller oject on the right (e.g., a building and a book).
]
Part D, Making Connections
2. More than three thousand years ago, sculptors of the Olmec culture on Mexico's Gulf Coast carved some of the most massive heads that have ever been created. Learn more about the Olmec heads and the people who carved them.
a. One of the heads at the Olmec site of San Lorenzo is shown from four angles. It is 6 ft tall . Estimate the width and depth of the head to the nearest foot. [width: 5 ft; depth: 3 ft]
b. Assume that the stone from which the head was carved was a rectangular prism with the dimensions in a. Sketch the prism showing cubic units measuring 1 ft x 1 ft x 1 ft.
c. What was the volume of the prism? [90 ft3]
d. Estimate the weight of the head, if each cubic foot of the material it is made from weighs 150 lb.
[Estimates will vary; about 13,000 lb]
Geometric designs appear in the works of artists as diverse as Native American weavers and window designers of the Middle Ages.
Part A, Triangles, Parallelograms, and Trapezoids
1. Using the Internet, answer the following questions.
a. Look at the Navajo "Gallup Throw Rug" . The design is made from three identical (or nearly identical) geometrical patterns. Draw a careful sketch of the middle of the three patterns.
b. Draw a horizontal line of symmetry through the middle of the pattern. Then identify each polygon in the figure using "T" for triangle, "TZ" for trapezoid, and "P" for parallelogram.
c. At the bottom of the pattern is a dark triangle with a base of about 2 in. and a height of about 1 in. On its top vertex balances a dark polygon. Find the area of the triangle. Then estimate the area of the polygon. Explain how you made your estimate.
d. Use triangles, parallelograms, and trapezoids to create a simple symmetrical pattern. Use at least two of each type of polygon in your pattern. Draw the line or lines of symmetry. [Patterns will vary.]
Part C, Making Connections
2. The Cathedral at Chartres, France, was built during the 12th Century. The cathedral's stained glass windows are masterpieces of medieval art. Study the Window at the main entrance to the cathedral.
a. The figure in the center circle of the window is life-size. Estimate the radius of the entire window.
[Estimates will vary. About 25 ft]b. Based on your estimate, what is the circumference of the window? What is the area? Use 3.14 for π.
[Answers will depend on estimates of radius; 157 ft; 1962.5 ft2]c. Around the outside of the window are 12 approximate semi-circles (blue in the figure). Each contains a circle (red). Estimate the diameter of each semi-circle. Explain how you made your estimate.
d. Estimate the diameter of each red circle.
[Red circle diameters are approximately 1/3 times blue circle diameters = 1/3 x 12 or about 4 ft.]e. Compare the diameter of each red circle with the diameter of the entire window.
Ancient peoples around the world used massive stones to build burial chambers. Some of the most extensively studied of these chambers are in England and in Egypt.
Part B, Prisms and Volume
1. Stonehenge is the most famous of the ancient sites in Great Britain, but there are many other such sites.
a. Find the measurements (ft) and estimated weight of "Lanyon Quoit". [17.5 ft long, 9 ft wide; 13.5 tons]
Part D, Making Connections
2. The three Great Pyramids at Giza, Egypt, are named for the Egyptian pharaohs Khufu, Khephren, and Menkaure. They were built more than 4000 years ago.
a. The Pyramid of Khufu has a square base. Sketch the pyramid, showing the length of each side ("the base") and the original height.
b. Find the quotient
. Round to the nearest hundredth. What famous number does this quotient approximate? [3.13; π]
c. Most people believe that the above number was not discovered until at least a thousand years after the Great Pyramids were built. A few people, however, argue that the quotient
[Menkaure, 3.27; Khephren, 3.00. Answers will vary. The quotients are not consistently near π. However, they're close. Perhaps the builders of the pyramids decided to show their knowledge of π only in the Khufu pyramid. Or perhaps the Khufu quotient is close to π only by coincidence. Students should see that the Khufu quotient presents interesting but by no means conclusive evidence that the Egyptians may have known about π.]d. The height of each face of the Khufu pyramid is 612 ft. Find the surface area of the pyramid (including the base).
[1,491,412 ft2]e. Here's another demonstration of advanced Egyptian mathematics--or perhaps just a coincidence: Find the square root of the area of one face of the Khufu pyramid. What do you notice about the result?
f. Find the volume of the Khufu pyramid.
[91,341,571 ft3]g. Record data on the weight and number of stones used to build the Khufu pyramid. Use your answer to f to find the volume of each stone and the density of each stone--the weight per cubic foot.
