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SQUARE ROOT OF FRACTIONS. 223.-51. Find the square root of a.
13 V3 V3 1.732+ V4
- .866+, Ans. VA
2 If the denominator is a perfect square and the numerator is not, divide the approximate square root of the numerator by the square root of the denominator.
52. Find the square root of k.
3 If the denominator is not a perfect square, reduce the fraction to an equivalent fraction having a perfect square for a denominator.
Find the square root of the following fractions to three places of decimals : 53. V 55. VTT
59. 14. 56. VI. 58. VED
APPLICATIONS OF SQUARE ROOT. 224. The Square. The side of a square is equal to the square root of its area.
WRITTEN EXERCISES. 1. A square farm contains 40 acres; how long is it?
2. What will it cost to fence a square field containing 10 acres at $1.20 a rod ?
3. It requires 1250 bricks, each 8 in. long and 4 in. wide, to cover a square pavement; how long is it?
4. Find the length of a square court which requires 6889 marble blocks, each 9 inches square, to cover the floor.
5. Find the dimensions of a rectangular field which is twice as long as it is wide, and contains 80 acres.
6. If it costs $144 to put a wire fence around a field 96 rods long and 24 rods wide, how much less will it cost to fence a square field of equal area ?
7. A general drew up his army of 4325 men in the form of a square, and had 100 men over; required the number of men on one side of the square.
8. A general drew up his army in the form of a square, and found he lacked 209 men to complete the square: he then diminished the side of the square by 3 men, and found he had 1000 men remaining; how many men in the army?
9. A square lake which contains 10 acres has a gravel walk 8 ft. wide around it; how many square yards in the walk ?
225. The Right Triangle.
A Right Triangle is a triangle which has one right angle.
226. The Hypotenuse is the side opposite the right angle. Either of the two sides forming the right angle may be taken as the Base, and the other side as the Perpendicular.
It is proved in Geometry that
In any right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Hence the square of either side equals the square of the hypotenuse diminished by the &quare of the other side.
WRITTEN EXERCISES. 1. The base of a right triangle is 24 inches, and the perpendicular is 18 inches; required the hypotenuse.
We have 182 + 24 = .7 324 + 576 = / 900 = 30. Hence the hypotenuse is 30 in., Ans. 2. The hypotenuse of a right triangle is 75 ft., and the base is 60 feet; required the perpendicular.
We have ✓754 – 602 ✓ 5625 – 3600 2025 = 45. ' Hence the perpendicular is 45 ft., Ans. 3. The hypotenuse of a right triangle is 50 ft., and the perpendicular is 30 ft.; required the base.
4. The base of a right triangle is 120 inches, and the hypotenuse is 40 ft.; required the perpendicular.
5. Find the distance between the opposite corners of a field 45 rods long and 20 rods wide.
6. A square farm contains 90 acres ; required the distance between the opposite corners.
7. B, standing 36 ft. from a tree, shoots a bird from its top; how high is the bird from the ground, if the ball travels 60 ft. and starts 6 ft. from the ground ?
8. A ladder 52 ft. long stands against a building; how far must it be drawn out at the bottom, that the top may be lowered 8 ft. ?
9. A room is 40 ft. long, 25 ft. wide, and 12 ft. high ; how far is it from an upper corner to the lower opposite corner ?
10. The diagonal of a square is 25 feet; find the approximate value of its side to three places of decimals.
11. A tree was broken off 24 ft. above the ground, and fell so that its top struck 32 feet from the foot of the tree, the end resting on the stump; what was the height of the tree?
12. The area of a square is 6 A. 64 P.; find the length of the diagonal.
227. Similar Figures.
Similar Figures are those which have the same form, but differ in area.
It is proved in Geometry that
1. The areas of similar figures are to each other squares of their like dimensions.
2. The like dimensions of similar figures are to each other as the
square roots of their areas.
Area A BCD: area EFGH length A B’ : length EF. Or, 64 16
4. Or, Area ABCD:area EFGH=length A B: length EF. ✓ 64
1. The area of a triangle whose base is 32 in. is 768 sq. in.; what is the base of a similar triangle whose area is 48 sq. in. ?
PROCESS. 768 : 48 = 322 : x2 Since the triangles are similar, their areas
are to each other as the squares of their 48 x 1024
64 bases; hence we have the proportion 768 : 768
48 = 322 : x'.
Solving the proportion, x = 8.
2. The length of a rectangular field containing 12 acres is 48 rods; what is the length of a similar field containing 48 acres ?
3. There are two circular lakes: one is 4 miles in diameter and the other 20 miles; the second is how many times as large as the first ?
be A has a circular garden 11 feet in diameter, and wishes to lay out another 16 times as large; what must be the diameter of the latter ?
5. B has a field 60 rods wide and 72 rods long, which contains 27 acres; required the dimensions of a similar field which contains 18% acres.
6. If it costs $60 to paint a house 24 ft. long and 20 ft. wide, what will it cost to paint with the same kind of color a house whose dimensions are three times as large ?
7. If a horse tied to a stake by a rope. 10.09 rd. long can graze upon 2 acres of land, how long must the rope be that he may graze upon 12 acres ?
CUBE ROOT. 228. By involution
1 1, 103 1000,
The cube of 1, the smallest number of one figure, is 1; the eube of 3 is 27; the cube of 9, the largest number of one figure, is 729; hence the cube of a number consisting of one figure contains one figure, two figures, or three figures.
The cube of 10, the smallest number consisting of two figures, is 1000; the cube of 30 is 27000; the cube of 99, the largest number of two figures, is 970299; hence the cube of a number of two figures contains four, five, or six figures; that is, three times two, or three times two less one or less two.