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QUESTIONS ILLUSTRATIVE OF THE APPLICATION OF ALGEBRA TO MENSURATION.

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represent? Find, first, the value of B' in terms of the remaining quantities; and second, the value of h' in terms of the remaining quantities. What does the formula represent when h'=0?

2. What does the formula V =

π h2

(3 D 2 h) represent ?

6

and find the value of D in terms of the remaining quantities.

3. What does the formula V = * h (3 63

πh b2
6 4

+ h2) represent ? and

find the value of b in terms of the remaining quantities,

and also the diameter of the sphere in terms of V and h.

4. What does the formula S =

b2
4

+h2 represent? and find

the value of b in terms of S and h, and also the diameter

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and find the values cf D and d in terms of the remaining quantities.

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7. Explain the formula S = π (D2 — d2), and find the value of

D when s and d are given quantities.

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value of d in terms of the remaining quantities.

10. What do the formula (1) C = π D, (2) A =

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π D2

Section 3.

11. The area of a circle is equal to (a) times the circumference; find the area.

12. The area and circumference of a circle are equal to (a) times the diameter; find the area.

13. The inscribed square of a circle is one-half the circumscribing square.

14. The square of the area of a circle is (a) times the circumference added to (b) times the diameter; find the area.

Section 4.

15. The area and circumference of a circle are equal to π ; find the area.

16. The diameters of two circles are D and (D+h), and their

corresponding areas A and A'; show that

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A' A

h

is

17. The difference between the circumference and area of a circle is equal to their quotient; find the area.

18. The sum of the circumference and area of a circle is equal to their product; find the area.

Section 5.

19. The area of a circle is equal to (a); find the diameter.

20. The diameter, circumference, and area of a circle are in geometrical progression; find the area.

21. The area of a circle is equal to its circumference added to (a); find the area.

Section 6.

22. The area of a triangle is (a), and the base is equal to (n) times the perpendicular; find the perpendicular.

23. When the circumference, diameter, and area of a circle are in geometrical progression, show that the area in square feet is equal to the diameter in lineal feet divided by π.

24. The circumference, area, and diameter of a circle are in arithmetical progression; find the area.

25. The area of an equilateral triangle is equal to its circumference; find the area.

Section 7.

26. The area of a triangle is equal to (n) times the perpendicular added to the base, and the base is equal to the per

pendicular; find the area.

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27. The base, perpendicular, and area of a triangle are in geometrical progression; find the area when the common ratio

is r.

28. The base, perpendicular, and area of a triangle are in arithmetical progression; find the area when the common difference is d.

Section 8.

29. The perpendicular, circumference, and area of an equilateral triangle are in geometrical progression; find the side.

30. The area of a concentric circle is equal to the sum of the inner and outer circumferences; find the diameters when their product is 21.

31. The sum of the base and perpendicular of a triangle is equal to its area; find the area when the ratio of the base to the perpendicular is equal to n.

32. The difference between the base and perpendicular of a of its area; find the perpendicu

triangle is equal to

(1)

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33. The area of a concentric circle is equal to (a); find the outer diameter when it is equal to (n) times the inner diameter.

34. The area of a concentric circle is equal to the difference between the diameters, and the ratio of the greater to the less diameter is ; find the area of the concentric circle.

35. The less diameter, greater diameter, and the area of a concentric circle are in arithmetical progression, whose common difference is d; find the area.

36. The less diameter, greater diameter, and the area of a concentric circle are in geometrical progression, whose common ratio is (r); find the area.

Section 10.

37. The volume of a concentric cylinder, whose base is a circle, is equal to the sum of the squares of the outer and inner surfaces; find the volume of the cylinder when d2 = 1 — 4 π L. L being its length.

38. The inner surface, outer surface, and volume of a concentric cylinder, whose base is a circle, are in geometrical progression, whose common ratio is r; find the volume whose length is L.

39. The inner surface, outer surface, and volume of a concentric cylinder, whose base is a circle, are in arithmetical progression, whose common difference is ; find the volume whose length is L feet.

Section 11.

40. The volume, diameter, and height of a right cone are in geometrical progression; find the volume of the cone when the common ratio is r.

41. The area of the base of a right cone, the height, and volume, are in arithmetical progression, whose common difference is ; find the volume.

42. The diameter of the base of a right cone, the slant height, and the height are in arithmetical progression; find the volume when the common difference is 8.

Section 12.

43. The volume of a sphere is equal to the sum of its surface and diameter; find the volume and surface.

44. The diameter, volume, and surface of a sphere are in geometrial progression; find the ratio and volume.

45. Divide the diameter of a sphere into two parts, so that the shell shall be equal to

part of the sphere.

n

Section 13.

46. The diameter, volume, and surface of a sphere are in arithmetical progression; find the volume and surface.

47. The volume of a segment of a sphere is equal to its surface; find the volume in terms of the height of the segment.

48. A person being elevated (h) feet above the earth's surface, show that the extent of surface which is visible is measured Th D2 2h+ D'

by

where D is the diameter of the earth. And if

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49. The diameter of the base, height, and volume of a segment of a sphere are in geometrical progression, whose ratio is (r); find the volume.

50. The diameter of the base, height, and surface of a segment of a sphere are in arithmetical progression, whose common difference is 8; find the surface.

51. If a person be elevated (h) yards, and can then see (S) square yards on the surface of the earth, show that the diameter of the earth in yards is determined from the formula

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52. If a person be elevated (h) yards, and can then see to a distance of (T) yards in every direction, show that the

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