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(17.1)
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Find the volume of (a) A cubic yard in cubic inches (b) A cubic meter in cubic centimeters (1 m 100 cm)
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1 Find, to the nearest cubic inch, the volume of a cube whose edge is (a) 3 in; (b) 42 in; (c) 7.5 in; (d) 0.3 ft; (e) 1 ft 2 in. (17.3)
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Find, to the nearest integer, the volume of (a) A rectangular solid of length 3 in, width
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(b) A prism having a height of 2 ft and a square base whose side is 3 yd (c) A pyramid having a height of 2 yd and a base whose area is 6.4 ft2
Find, to the nearest integer, the volume of (a) A sphere with radius of 6 m (b) A cylinder having a height of 10 ft and a base whose radius is 2 yd (c) A cone having a height of 3 yd and a base whose radius is 1.4 ft
(17.5)
From V 3Bh, the volume formula for a pyramid or cone, derive volume formulas for each of the solids in Fig. 17-32. (17.6)
CHAPTER 17 Extending Plane Geometry into Solid Geometry
h h B B (a) r Cone s B
h w B l (c) Pyramid with rectangular base
s (b) Pyramid with square base
(d) Cone where h = 2r
Fig. 17-32
Find a formula for the volume of each solid in Fig. 17-33.
(17.7)
h e e e e e (a) e h l (b) w r
Fig. 17-33
Transformations
18.1 Introduction to Transformations
Two figures are congruent if one can be moved so that it exactly overlaps the other. A figure cut out of paper can be turned, slid, and flipped over to see if it matches up with another figure. If the figure is put on a graph, then these movements will change the coordinates of the points. A transformation is a way to describe such a change of coordinates.
18.2 Transformation Notation
A transformation begins with a general description of a point, such as P(x, y) which represents a point P with coordinates x and y. Following this is an arrow A and then a description of the point s image, the place where it ends up after the move. Usually the image of P is called P, the image of A is called A , and so on. For example, the transformation P(x, y) A Pr(x 5, 4 y) means that the point A(2, 1) is moved to Ar(2 5, 4 1) Ar( 3, 3), the point B(3, 5) is moved to Br(3 5, 4 5) Br( 2, 1), and the point C(6, 1) is moved to Cr(6 5, 4 1) Cr(1, 3). This transformation flips the triangle ^ABC over and slides it to the left, as shown in Fig. 18-1.
Fig. 18-1
SOLVED PROBLEMS
18.1 Using transformation notation Name the image of the points A(3, 1), B(3, 4), and C(5, 1) under the following transformations: (a) P(x, y) A Pr(x (b) Q(x, y) A Qr(x 2, y 5, y) 1)
(c) R(x, y) A Rr(5x, 5y)
(d) S(x, y) A Sr( y, x) (e) T(x, y) A Tr(y, 5
Solutions
(a) Ar(3 (b) A (3 2, 1 5, 1) 1) Ar(5, 0), Br(3 5, 4) 2, 4 1)
CHAPTER 18 Transformations
Br(5, 3), and Cr(5 5, 1)
2, 1 C (10, 1)
Cr(7, 0)
A (8, 1), Br(3
Br(8, 4), and C (5
(c) Ar(5 # 3, 5 # 1)
Ar(15, 5), Br(5 # 3, 5 # 4)
Br(15, 20), and Cr(5 # 5, 5 # 1)
Cr(25, 5)
(d) Ar( 1, 3), Br( 4, 3), and Cr( 1, 5) (e) Ar(1, 5 3) Ar(1, 2), Br(4, 5 3) Br(4, 2), and Cr(1, 5 5) Cr(1, 0)
18.3 Translations
A transformation that slides figures without flipping or rotating them is called a translation. The translation that slides everything to the right a units and up b units is P(x, y) A Pr(x a, y b).
SOLVED PROBLEMS
Performing a translation Let rectangle ABCD be formed by A( 1, 4), B( 1, 3), C(3, 3), and D(3, 4). Graph rectangle ABCD and its image under the following translations: (a) P(x, y) A Pr(x (b) P(x, y) A Ps(x (c) P(x, y) A P-(x
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