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Let A be the centre of the sphere, B the point from which straight lines are drawn, BC one of the straight lines, meeting the sphere at C. Draw CD perpendicular to AB. Let E denote the point where AB cuts DEB the sphere. Then the surface of which we have to find the area is the curved surface of the segment of which DE is the height. shall now find the length of DE.

A

Taking all the lengths in feet we have

AB=15, AC=AE=12;

We

hence, by Art. 60, we obtain BC=9: and, by Art. 151, we

36
5

shall find that AD= =72. By Art. 60 we shall see that AD is the square root of 144-51 84, that is the square root of 92:16; it will be found that this is 9'6. We may obtain this result more easily by similar triangles; by Art. 37 we have

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Therefore the area of the curved surface of the segment in square feet 24 × 2 × 12 × 3·1416=180·95616.

An eye placed at B would see exactly that portion of the surface of the sphere of which we have just found the

area.

If we wish to know merely what fractional part of the whole surface of the sphere is visible to an eye placed at B, we have only to form the fraction which has the length of DE for the numerator, and the length of the diameter of the sphere for denominator: in the present case the fraction is

2.4
24'

that is

10

EXAMPLES. XXXIX.

Find the areas of curved surfaces of segments of a sphere having the following dimensions:

1. Height of segment 10 inches, circumference of sphere 85 inches.

2. Height of segment 2 feet, circumference of sphere 20 feet.

3. Height of segment 9 inches, radius of sphere 16 inches.

4. Height of segment 24 feet, radius of sphere 3.25 feet.

Find the areas of the whole surfaces of segments of a sphere having the following dimensions:

5. Height of segment 8 inches, radius of sphere 25 inches.

6. Height of segment 2 feet, radius of sphere 7 feet. 7. Height of segment 11 inches, circumference of sphere 90 inches.

8. Height of segment 3 feet, circumference of sphere 27 feet.

Find the areas of the whole surfaces of zones which are the differences of segments having the following dimensions:

9. Radius of sphere 15 inches, heights 6 and 9 inches.

10. Radius of sphere 11 feet, heights 3 feet and 10 feet.

11. Circumference of sphere 75 inches, heights 3 and 7 inches.

12.

Circumference of sphere 40 feet, heights 2 feet

and 5 feet.

Find the areas of the whole surfaces of zones of a sphere having the following dimensions.

13. Radius of sphere 16 inches; distances of ends of zone from the centre 5 inches and 9 inches, on the same side of the centre.

14. Radius of sphere 9 feet; distance of ends of zone from the centre 2 feet and 3 feet, on opposite sides of the centre.

15. Circumference of sphere 90 inches; distances of ends of the zone from the centre 6 inches and 10 inches, on the same side of the centre.

16. Circumference of sphere 32 feet; distances of ends of the zone from the centre 3 feet and 4 feet on opposite sides of the centre.

17. A sphere is 80 feet in diameter: find what fraction of the whole surface will be visible to an eye placed at a distance of 41 feet from the centre.

18. A sphere is 90 feet in diameter: find what fraction of the whole surface will be visible to an eye placed at a distance of 8 feet from the surface.

19. Find at what distance from the surface of a sphere an eye must be placed to see one-sixteenth of the surface.

20. Find at what distance from the surface of a sphere an eye must be placed to see one-eighth of the surface.

221

SIXTH SECTION.

PRACTICAL

APPLICATIONS.

XL. INTRODUCTION.

358. WE have already given various applications of the Rules of Mensuration to matters of practical interest; for instance the Examples of the calculations of the expense of carpeting the floors of rooms and papering the walls which occur under Chapter XI. Such applications require only a knowledge of the elements of Arithmetic in addition to the principles of Mensuration already explained.

A few more examples will be given at the end of the present Chapter.

359. There are however other applications which require the student to know the meaning of certain technical terms, or to employ certain approximate Rules admitted by custom. We shall consider these cases in the three Chapters which immediately follow the present Chapter.

EXAMPLES. XL.

1. A room is 24 feet 3 inches long, 11 feet 9 inches broad, and 11 feet 6 inches high: find the cost of painting the walls at 1s. 6d. per square foot.

2. Find the cost of painting the four walls of a room which is 32 feet 4 inches long, 15 feet 8 inches broad, and 11 feet 6 inches high at 3 shillings per square foot.

3. Find the cost of painting the four walls of a room whose length is 24 feet 3 inches, breadth 15 feet 8 inches, and height 11 feet 6 inches at 4 shillings per square foot.

4. A room is 24 feet 10 inches long, 16 feet broad, and 12 feet 4 inches high: find the cost of painting the four walls at 9 pence per square foot.

5. The length of a room is 7 yards 1 foot 3 inches, the breadth is 5 yards 2 feet 9 inches, and the height 4 yards 6 inches: find the cost of papering the walls, supposing the paper to be a yard broad and to cost 9d. per yard.

6. A cubical box is covered with sheet lead which weighs 4lbs. per square foot; and 294 lbs. of lead are used: find the size of the box.

7. Find the cost of lining the sides and the bottom of a rectangular cistern 12 feet 9 inches long, 8 feet 3 inches broad, 6 feet 6 inches deep with sheet lead which costs £1. 8s. per cwt., and weighs 8 lbs. to the square foot.

8. A cistern open at the top is to be lined with sheet lead which weighs 6 lbs. to the square foot; the cistern is 4 feet 6 inches long, 2 feet 8 inches wide, and holds 42 cubic feet: find the weight of lead required.

9. A box with a lid is made of planking 14 inches thick; if the external dimensions be 3 feet 6 inches, 2 feet 6 inches, and 1 foot 9 inches, find exactly how many square feet of planking are used in the construction.

10. A flat roof is 17 feet 4 inches long and 13 feet 4 inches wide; find the cost of covering it with sheet lead one-sixteenth of an inch thick, supposing that a cubic inch of lead weighs 63 ounces avoirdupois, and that 1 lb. of it costs 3 d.

11. Find the cost of painting the wall of a cylindrical room 16 feet high, and 18 feet in diameter, at 7 d. per square yard.

12. Find the cost of painting a conical spire 64 feet in circumference at the base, and 108 feet in slant height at 7. per square yard.

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