Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

98. The radii of the bases of a frustum of a right circular cone are 11" and 12.5" respectively, and the height of the frustum is 5". In the original cone, find the area of the curve surface, the area of the total surface, the altitude, and the volume.

99. The area of the total surface of a polyhedron weighing 64 lb. is 340 sq. in. Find the surface of a similar polyhedron made of the same material and weighing 1000 lb.

100. The bottom and top diameters of a tub are to be 24" and 30" respectively. What depth should be allowed so that the tub shall hold 30 gal. (1 gal. = 231 cu. in.)?

101. A pump has a cylindric barrel 4" in diameter, and the volume of water in 1' of the length is pumped at each stroke. Find the number of strokes necessary to fill a tub in the form of a frustum of a cone 3' in diameter at the bottom, 4' in diameter at the top, and 1'6" high.

102. It is desired to double the capacity of a cylindric boiler, but to keep the diameter and height in the same ratio. By what per cent will the total area be increased?

103. If the area of the surface of a spherical balloon is doubled, by what per cent is the circumference increased? the diameter? the volume? the radius?

104. Find the least amount of wood which it is necessary to waste in cutting a cube out of a wooden sphere 4" in diameter.

105. Find the volume of the largest sphere that can be cut from a cone of revolution 14" high and 12" in diameter.

106. The specifications for a brass sphere 1" in diameter require that the alloy contain one part of zinc to two parts of copper by volume. Given that 1 cu. ft. of copper weighs 550 lb. and that 1 cu. ft. of zinc weighs 428 lb., find the weight of the sphere.

107. The wooden part of a top consists of a conic frustum with a hemispherical end. The greatest and least diameters of the frustum are 3.5" and 0.5", and the slant height is 3.5". Find the total volume.

108. If an iron sphere 4" in diameter is placed in a conic vessel which is full of water and whose altitude and diameter are each 5", how much water will run over?

109. Find the volume of the largest sphere that can be cut from a metal cone whose base has a diameter of 7" and whose slant height is 7".

110. If a circular hole 1" in diameter is bored through a sphere 2" in diameter and the axis of the hole passes through the center of the sphere, what is the volume of the part of the sphere that is left?

111. A cylinder which is 24" in diameter and 16" in height is inscribed in a sphere. Find the area and the volume of the sphere.

112. Prove that the volume of a regular octahedron is 0.47s, approximately, and that the area of the total surface is 3.46 s, approximately, where s is an edge of the solid. From these results find the area and the volume of a regular octahedron 3" on an edge.

113. A stone post is to be surmounted by a sphere 14" in diameter, and in order to give the sphere a base upon which to stand a segment of one base 2" high is cut from the sphere. Find the volume of the segment of the sphere that is placed on the post.

114. If the exposed surface of the segment of the sphere in Ex.113 is polished, how many square inches are polished?

115. Find the volume of a spherical sector of a sphere of radius r, given that the area of the zone which forms the base of the spherical sector is 4 Tr2. Draw the figure.

V. RECREATIONS

267. Fallacies. Below are given a few curious problems and interesting fallacies, generally based upon incorrect constructions or statements, which should be undertaken, if time allows, simply as recreations.

1. Any point on a line bisects it.

In the figure below let BC be any line and P any point on it.
Construct an isosceles AABC upon BC as base,

and draw AP.

ZB=LC,

A

AABP is congruent to AACP.

Since

and

AB= AC,

AP=AP,

then

Hence

BP=PC,

[blocks in formation]
[blocks in formation]

Let ABC be any ▲ in which AC is not equal to BC.

Bisect C and construct the 1 bisector of AB, letting it meet the bisector of C at P. They must meet, for if they were II, the bisector of C would be to AB and hence would bisect it, thus coinciding with the bisector MP. This would be possible only if AC=BC, which is contrary to what is assumed above.

Draw PDL to AC and PE 1 to BC.

Then, since CP bisects ≤C, we have PD = PE;

and since MP is the bisector of AB, then AP=BP.

[ocr errors]

E

[blocks in formation]

Then AAPD is congruent to ABPE, and hence AD = BE.
Similarly, ▲PDC is congruent to ▲PEC, and hence DC = EC.
Adding, AD+DC=BE+EC, or AC = BC.

Hence every ▲ is isosceles.

3. Find the area of this triangle to the nearest 0.1 sq. ft.

You may use the second formula for the area of

a triangle given in § 277; or, if you prefer, draw the figure to scale, measure the altitude, and then find the area by means of § 25, 3.

79

148

4. If A > B and B> C, then it follows that A=B=C.

In this figure the arcs of the © are all tangent to PT at P. Then, if A=XPT, B=ZYPT, and C=ZZPT,

A > B > C.

Now in plane geometry we define the Zbetween two as the between their

tangents at a common point.

P

-T

But the between the tangents at P of any two of these is 0 and hence

A=B=C.

5. Construct a triangle such that the sum of the interior angles is less than 180°.

The three of which the arcs are here shown

A

are tangent at A, B, C.

Then, as in Ex. 4, the

between the tangents

at a common point of any two is 0.

B

Hence the sum of the 4 of the ▲ formed by the tangents at A, B, C is 0.

6. All circles, however large, have equal circumferences.

Let two of unequal radii

AP and AQ be fastened together, and let them roll along from A to A'.

Then P reaches P' when Q

reaches Q'.

P

P

Since the have rolled along equal distances, their circumferences must be equal.

7. Two coins A and B of the same size are placed upon a table so that A is tangent to B. If B is kept fixed and A is rolled around B, always remaining tangent to B, how many revolutions does A make in rolling once around B?

Play fairly; give your answer and reason for it before experimenting. 8. A man who had a window 2 ft. wide and 4 ft. high wished to double its area. He did so, and still the window was only 2 ft. wide and 4 ft. high. How was this possible?

9. The sum of the parallel sides of a trapezoid is zero. In the figure below let ABCD be the trapezoid with bases AB (or b) and CD (or a).

and

Now let DC be produced to S and BA to P so that CS = b and AP = a. APAQ is similar to ASCQ,

Then

[blocks in formation]
[merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[ocr errors]
[ocr errors]

whence

[merged small][merged small][ocr errors]

But

=

-1.

[blocks in formation]
[ocr errors][merged small][merged small]

a = - b ; a+b=0.

10. Any number, however large, is equal to zero.

On a piece of squared paper mark out a square which shall be 8 by 8, and then draw lines dividing it into three parts A, B, C, as shown. Then mark out

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

small squares in the large square is 8 × 8, or 64, and the number of small squares in the☐ is 5 × 13, or 65.

[blocks in formation]

Multiplying these equals by any number, say 25, we have

and hence

25=0,

any number is equal to zero.

« ΠροηγούμενηΣυνέχεια »