100. The floor of a room whose length is AB is covered with regular hexagonal tiles. How many tiles are in one row that extends over the entire length (AB) of the room, if each side of the hexagons equal 2 in., AB = 10 ft., and the tiles are placed in the position indicated in the figure?

101. The approximate value of the circumference of a circle is sometimes found by carpenters · as follows. Draw the equilateral triangle AOB and produce its altitude OD to E. The circumference equals 6 AO+2 DE. Find the error if π is equal to 3.1416.

102. Two wheels A and B are connected by a belt, and their radii are respectively 2 ft. and 9 ft. If the larger wheel (B) makes 40 revolutions per minute, how many revolutions per minute will the smaller wheel make?

103. In the figure of the preceding exercise find the length of the belt if AB = 14 ft.

A

(Radii are 2 ft. and 9 ft.)

104. The gauge of an automobile (i.e. the distance between two wheels on the same axis) equals 4 ft. 81⁄2 in. How many inches more than an inner wheel does an outer wheel travel when the car turns around a rectangular corner?

105. The circumference of the equator is (approximately) 25,000 mi. Imagine a concentric circle

whose circumference is 1 ft. longer. What is the difference of the radii of the two circles?

106. The annexed figure represents a small wire fence used to protect flower beds, etc. How many feet of wire are needed per running foot of fence if AB = 1 ft., CD = 9 in., DE 3 in.?

=

Ex. 106

107. If two streets meet at an angle of 120°, and an automobile turns from one into the other, what is the difference between the dis

tances traveled by an outer and an inner wheel of an automobile? (Gauge 4 ft. 8 in.)

108. The span of a circular arch AB equals 10 ft., and its height CD=2 ft. Find the radius of the arch.

D

=

109. If the earth's orbit be assumed to be a circle whose radius 93,000,000 mi., and the year 365 da., how many miles per hour does the earth travel?

=

110. Find the area of an equilateral Gothic arch ABC, if AC:

= 4 ft.

111. In the same figure, what is the area of each of the smaller equilateral arches if AD = DC 2 ft.?

B

A

D

112. In the same diagram, what is the area of the circle that touches the three equilateral arches?

113. A sidewalk 5 ft. wide turns a corner. Find the area of the curved portion of the sidewalk if the radius AB=8 ft. and ZA = 60°.

115. How many miles per hour does a point whose latitude equals 45° travel in consequence of the earth's rotation about its axis P if we assume the diameter of the earth equal to 8000 mi., and the time of one rotation equal to 23 hr. 56 min.?

116. It is found that a water pipe whose diameter is 2 in. supplies half the amount of water that is needed in a building. What would be the diameter of a pipe that would supply the entire required amount of water?

117. What should be the diameter of a water pipe that supplies the same amount of water as both an 8-in. and a 6-in. pipe, if we assume that all other conditions, as pressure, etc., are the same in the three pipes?

118. A rail 50 ft. long must be bent through what angle (i.e. O formed by perpendiculars at its end) if the radius of curvature equals 360 ft.?

119. In laying a curved track, a rail 55 ft. long is bent through an angle of 17° 30'. Find the radius of curvature.

120. If O is the center of arc AB, BC 1 OA, and 20 is small, then the difference between AB and BC is very small. If we have to find the value of BC, we may, in such cases, find the length of arc AB instead. If 0=4°, the error due to this simplification equals .00005 of the result, and for smaller angles the error is much smaller, since it is approximately proportional to the cube of the angle.

C

A

If BC is a tower standing on a horizontal plane A O, BO = 2000 ft. and 40 = 2°. Find the height of the tower.

121. The apparent diameter of the

moon equals 30', i.e. a line EM drawn from earth (E) to the center of the moon (M) and a tangent EA include an angle of 15'. If EM=243,000 mi., find the radius of the moon.

E

M

122. The apparent diameter of the sun is approximately the same as that of the moon (preceding problem). What is the diameter of the sun if its distance from the earth equals 93,000,000 mi.?

SOLID GEOMETRY

BOOK VI

LINES AND PLANES IN SPACE - POLYHEDRAL

ANGLES

477. DEF. Geometry of space or solid geometry treats of fig ures whose elements are not all in the same plane.

(17)

478. DEF. A plane is a surface such that a straight line joining any two points in it lies entirely in the surface. (14)

A plane is determined by given points or lines, if one plane and only one, can be drawn through these points or lines.

PROPOSITION I. THEOREM

479. A plane is determined:

(1) By a straight line and a point without the line. (2) By three points not in the same straight line. (3) By two intersecting straight lines.

(4) By two parallel straight lines.

(1) To prove that a plane is determined by a given straight line AB and a given point C.

Turn any plane EF passing through AB about AB as an axis until it contains C.

If the plane, so obtained, be turned in either direction. would no longer contain C.

Hence, the plane is determined by AB and C.

(2) To prove that three given points 4, B, and C determine a plane.

Draw AB.

Then AB and C determine the plane. (Case 1.) (3) To prove that two intersecting straight lines AB and AC determine a plane.

[Proof by the student.]

(4) To prove that two parallel lines AB and CD determine a plane.

The parallel lines AB and CD lie in the same plane by definition.

Since AB and the point c determine a plane, the two parallels determine a plane.

A

B

480. COR. The intersection of two planes is a straight line. For the intersection cannot contain three points not in a straight line, since only one plane can be passed through three such points.

NOTE. Constructions in solid geometry are usually treated as if we possessed the physical means of making a plane that is determined by any of the above elements. Obviously this could be done only if we employed models. With ruler and compasses the work is largely symbolic; i.e. we obtain pictures which are symbols of the figures to be constructed, but we do not always obtain the required figures themselves, as we do in plane geometry.

Ex. 1438. Why is a photographer's camera or a surveyor's transit supported by three legs?