Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide

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multiples of 108 and that are numbered according to their angle

counterclockwise from due east. You check out of Motel Delta

and ask the clerk how to get to State Complex Epsilon. The atten-

dant says, ‘‘Just follow this avenue outward 2 blocks, then turn

left and travel counterclockwise exactly 9 blocks.’’ How far must

you drive?

(a) It is impossible to answer this question without more infor-

mation.

(b) 0.9 km

(d) 1.8 km

(e) 2.2 km

21. A major league baseball inﬁeld is built to speciﬁc dimensions. Home

plate, ﬁrst base, second base, and third base are all at the vertices of a

square that measures 90 feet (90 ft) on a side. The diagonal running

from home plate to second base is the same length as the diagonal

running from ﬁrst base to third base. Let P be the point where

these diagonals intersect. The pitcher’s rubber lies along the diagonal

from home plate to second base, and is 60.5 ft from home plate.

Which of the following statements is true?

(a) The pitcher’s rubber is closer to home plate than to second base.

(b) The pitcher’s rubber is closer to second base than to home plate.

base.

(d) The pitcher’s rubber is centered at point P.

(e) None of the above.

22. Suppose you see the shadow of a post cast by the sun on a ﬂat pave-

ment. The cosine of the sun’s angle above the horizon is equal to

(a) the height of the post, divided by the length of the post’s shadow

(b) the length of the post’s shadow, divided by the height of the post

post to the tip of its shadow

(d) the distance from the top of the post to the tip of its shadow,

divided by the height of the post

(e) none of the above

23. According to Fig. Test 3-3, point A suggests that

(a) sin 908 ¼ 1

(b) sin 1 ¼908

Test: Part 3 301

(d) sin 908 ¼1

(e) the equation of the graph is linear

24. Which of the following four statements (a), (b), (c), or (d), if any,

does Fig. Test 3-3 fail to suggest?

(a) sin 908 ¼sin þ908

(b) sin þ908 ¼sin 908

(d) sin þ308 ¼ 0.5

(e) The ﬁgure suggests that (a), (b), (c), and (d) are all true.

25. The extent of the domain of angles  shown in Fig. Test 3-3, in

radians, is

(a) –p/4   þp/4

(b) –p/2   þp/2

(d) 2p   þ2p

(e) 4p   þ4p

26. The graph of Fig. Test 3-3 suggests that the range of the sine func-

tion, when the independent variable is restricted to angles between

and including 1808 and þ1808,is

(a) 1  sin   1

(b) 0  sin   1

(d) 0  sin  1

(e) 1  sin  1

Fig. Test 3-3. Illustration for Part Three Test Questions 23 through 28.

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302

27. The curve shown in Fig. Test 3-3 is known as

(a) a sinusoid

(b) a circular wave

(d) a radial wave

(e) a Cartesian wave

28. The curve shown in Fig. Test 3-3 represents

(a) 1/4 of a wave cycle

(b) 1/2 of a wave cycle

(d) two full wave cycles

(e) four full wave cycles

29. Conventional time is measured according to the mean solar day

(MSD). For an observer in the northern hemisphere, the MSD is

the average length of time it takes for the sun to ‘‘revolve once

around’’ from a point directly in the south to a point directly in

the south on the following day. This is, by convention, divided

into 24 hours. This means that in 1 hour:

(a) the earth rotates 18 on its axis with respect to the sun

(b) the earth rotates 108 on its axis with respect to the sun

(d) the earth rotates 248 on its axis with respect to the sun

(e) the earth rotates 308 on its axis with respect to the sun

30. Imagine a city laid out on a level expanse of prairie, with ‘‘streets’’

that are concentric circles spaced 200 m apart and ‘‘avenues’’

that go radially outward from the city center at angles that are all

multiples of 108 and that are numbered according to their angle

counterclockwise from due east. You have been telling your

senator at State Complex Epsilon how much you hate her economic

policy, and then you ask her how to get to Tera Mall Theta. She

says, ‘‘Just follow this avenue outward 6 blocks.’’ What is the

straight-line distance between State Complex Epsilon and Tera

Mall Theta?

(a) It is impossible to answer this question without more infor-

mation.

(b) 0.6 km

(d) 1.8 km

(e) 3.6 km

Test: Part 3 303

31. The Cartesian plane is an example of a

(a) rectangular coordinate system

(b) polar coordinate system

(d) circular coordinate system

(e) spherical coordinate system

32. In Fig. Test 3-4, suppose that each of the faces is a parallelogram,

and s

¼ s

. In this case, we can be certain that the ﬁgure is a

(a) cube

(b) skew prism

(d) hexagonal prism

(e) parallelepiped

33. In Fig. Test 3-4, suppose angles x, y, and z each measure 908. Also

suppose that each of the faces is a parallelogram. In this case, we

can be certain that the ﬁgure is a

(a) cube

(b) skew prism

(d) hexagonal prism

(e) rhombus

34. In Fig. Test 3-4, suppose angles x, y, and z each measure 708. Also

suppose that each of the faces is a parallelogram. In this case, we

can be certain that the ﬁgure is a

(a) parallelepiped

(b) rhombus

Fig. Test 3-4. Illustration for Part Three Test Questions 32 through 35.

PART 3 Shapes and Places

304

(d) hexagonal prism

(e) skew prism

35. In Fig. Test 3-4, suppose h ¼ 5 m and the area of the base is 18 m

Also suppose that each of the faces is a parallelogram. What is the

volume of the ﬁgure?

(a) 23 m

(b) 30 m

(d) 90 m

(e) It is impossible to determine this without more information.

36. Imagine that you have two spherical balls, called Ball A and Ball B.

Suppose Ball B has four times the surface area of Ball A. What can

we say about their relative diameters?

(a) The diameter of Ball B is twice the diameter of Ball A.

(b) The diameter of Ball B is four times the diameter of Ball A.

(d) The diameter of Ball B is 16 times the diameter of Ball A.

(e) Nothing, without more information.

37. Suppose you are in a perfectly cubical room measuring exactly 3 m

high by 3 m wide by 3 m deep. You draw a straight line L along

one wall (call it wall X) so that line L runs ‘‘kitty-corner’’ diagonally

across the square formed by wall X, dividing wall X into two

identical triangles. What is the length, in meters, of line L?

(a) The square root of 3.

(b) The square root of 6.

(d) The square root of 18.

(e) The square root of 27.

38. Suppose you are in a perfectly cubical room measuring exactly

3 m high by 3 m wide by 3 m deep. You draw a straight line L

along one wall (call it wall X) so that line L runs ‘‘kitty-corner’’

diagonally across the square formed by wall X, dividing wall X

into two identical triangles. What is the area of either one of these

triangles?

(a) It is impossible to determine this without more information.

(b) 4.5 m

(d) 9 m

(e) 27 m

Test: Part 3 305

39. Suppose you are in a perfectly cubical room measuring exactly 3 m

high by 3 m wide by 3 m deep. You draw a straight line L along

one wall (call it wall X) so that line L is parallel to the ﬂoor, and

so that one point on line L is 2 m above the ﬂoor. From this, you

can be sure that line L

(a) intersects the ceiling along wall X, but not necessarily at a right

angle

(b) intersects the ceiling along some wall other than X, and not

necessarily at a right angle

(d) intersects the ceiling along some wall other than X at a right

angle

(e) does not intersect the ceiling

40. Suppose you are in a perfectly cubical room measuring exactly 3 m

high by 3 m wide by 3 m deep. You draw a straight line L along

one wall (call it wall X) so that line L intersects the ﬂoor at a right

angle. From this, you can be sure that line L

(a) intersects the ceiling along wall X, but not necessarily at a right

angle

(b) intersects the ceiling along some wall other than X, and not

necessarily at a right angle

(d) intersects the ceiling along some wall other than X at a right

angle

(e) does not intersect the ceiling

PART 3 Shapes and Places

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PART FOUR

Math in Science

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CHAPTER

Vectors and 3D

Cartesian three-space, also called rectangular three-space or xyz-space,is

deﬁned by three number lines that intersect at a common origin point. At

the origin, each of the three number lines is perpendicular to the other

two. This makes it possible to pictorially relate one variable to another.

Most three-dimensional (3D) graphs look line lines, curves, or surfaces.

Renditions are enhanced by computer graphics programs.

You should know middle-school algebra to understand this chapter.

Vectors in the Cartesian Plane

A vector is a mathematical expression for a quantity with two independent

properties: magnitude and direction. The direction, also called orientation,is

deﬁned in the sense of a ray, so it ‘‘points’’ somewhere. Vectors are used

to represent physical variables such as distance, velocity, and acceleration.

Conventionally, vectors are denoted by boldface letters of the alphabet.

309

In the xy-plane, vectors a and b can be illustrated as rays from the origin (0,0)

to points (x

, y

) and (x

, y

) as shown in Fig. 13-1.

EQUIVALENT VECTORS

Occasionally, a vector is expressed in a form that begins at a point other than

the origin (0,0). In order for the following formulas to hold, such a vector

must be reduced to standard form, so it ‘‘begins’’ at the origin. This can be

accomplished by subtracting the coordinates (x

, y

) of the starting point

from the coordinates of the end point (x

, y

). For example, if a vector a*

starts at (3,2) and ends at (1,3), it reduces to an equivalent vector a in

standard form:

a ¼fð1  3Þ, ½3 ð2Þg

¼ð2,  1Þ

Any vector a* that is parallel to a, and that has the same length and the

same direction (or orientation) as a, is equal to vector a. A vector is deﬁned

solely on the basis of its magnitude and its direction (or orientation). Neither

of these two properties depends on the location of the end point.

MAGNITUDE

The magnitude (also called the length, intensity,orabsolute value) of vector a,

written |a|ora, can be found in the Cartesian plane by using a distance

Fig. 13-1. Two vectors in the Cartesian plane. They are added using the

‘‘parallelogram method.’’

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