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(14) A man undertook to run round a square field containing 13 acres 1089 yards in 4 minutes. How many yards per minute must he run to do it?

(15) A square field contains 175056 sq. yds. Find the cost of fencing all round at 74d. per foot.

(16) A square plot of land is 55 links in length; what is the side of another plot of the same shape containing four times as much land?

(17) The sides of two squares contain 332 yds. and 249 yds. respectively; find the side of a square whose area is equal to the sum of the areas of the two squares.

(18) The sides of two squares measure 77 yds. 1 ft. 9 in. and 7 yds. 2 ft. 4 in. respectively; find the side of a third square whose area is equal to the sum of the areas of the first and second.

(19) The sides of three squares being 51%, 9, and 16 feet, find the side of the square which is equal to the sum of the three.

(20) The sides of two squares being 30 and 50, find the side of a square whose area is equal to the difference between the areas of the two squares.

(21) The perimeter of one square is 20 yds. and that of another 48 yds.; find the perimeter of a square which is equal in area to the two.

48. To find the side of a square when the value of its surface is given.

RULE. 1°. Divide the value of the whole surface by the value of a square unit, and the quotient will be the number of square units in the area.

2o. Extract the square root of this number, and the result will give the number of linear units in the side.

Ex. The cost of covering the floor of a square room with oilcloth at 4d. per sq. ft. was £5 8s.; find the length of the

room.

£5 88.1296d.

No. sq. units in room=1296÷4=324.

Length of side=sq. root of 324=18 ft.

Ex. XVI.

(1) A square garden was sold for £183 15s. If the land was valued at 3s. per sq. yd., find the length of the garden.

(2) Find the length of the side of a square enclosure, the paving of which cost £34 4s. 6d. at 6d. per square yard.

(3) Determine the side of a square field which cost £57 15s. 2 d. trenching, at 2d. per square yard.

(4) The rent of a square field at £2 14s. 6d. per acre amounts to £27 58.; find the cost of putting a paling round the field at 9d. per yard.

(5) Find the side of a square courtyard the expense of paving which is £21 3s. 6d. at 3s. 6d. per square yard.

THE RECTANGLE.

A Rectangle, or Oblong, is a parallelogram which has its opposite sides equal, and all its angles right angles.

CD and EF are called the length, or base.

CF and DE are called the breadth, or

height.

E

[blocks in formation]

F

50. To find the area of a rectangle, or oblong, when the length and breadth are given.

RULE. Multiply the length by the breadth, and the product will be the area.

Suppose we have a rectangle which is 4 in, long and 3 in. broad. Draw straight lines, an inch apart, parallel to the sides. The rectangle is thus divided into 12 equal figures, each of which is a square an inch long and an inch broad.

1

1

2

3

2

3

4

Such a square is called a square inch, and the rectangle containing 12 of them, its area is said to be 12 sq. in. The number 12 is the product of the numbers 4 and 3, which denote respectively the length and breadth of the rectangle; hence the rule.

Ex. 1. Find the area of the rectangle C D E F, when CD =27, and DE=11.

Area CDEF=27×11=297.

Ex. 2. What is the superficial content of a floor whose length is 5 ft. 3 in., and breadth 3 ft. 6 in.

(i.) By Reduction. 5 ft. 3 in. =63 in.; 3 ft. 6 in. =42 in.
.. Area 63 x 42=2646 sq. in. =18 sq. ft. 54 sq. in.
(ii.) By Fractions. 5 ft. 3 in. =5 ft.; 3 ft. 6 in. =33 ft.
.. Area 51 × 31 = 4×7 = 147 = 183 sq. ft.

=

54 sq. in.

=

=

=

18 sq. ft.

= 3.5 ft. 18 sq. ft. 54 sq. in.

(iii.) By Decimals. 5 ft. 3 in.=5.25 ft.; 3 ft. 6 in.
.. Area 5.25 × 3·5 = 18.375 sq. ft.
(iv.) By Duodecimals. 5 ft. 3 in. x3 ft.
18 sq. ft. 54 sq. in.

6 in.

=

18 sq. ft. 4' 6'

Ex. 3. Required the acreage of a rectangular field CDEF, whose adjacent sides CD, C F measure 8 ch. 25 lks. and 4 ch. 75 lks. respectively.

8 ch. 25. lks. = 825 lks.; 4 ch. 75 lks. =475 lks.

.. Area 825 × 475 = 391875 sq. lks. = 3·9187 5ac. = 3 ac. 3 ro. 27 po.

Ex. XVII.

Find the area in square feet of rectangles having the follow

[blocks in formation]

Find the area in square yards and feet of rectangles having the following dimensions:

(13) 3 yds. 2 ft by 8 yds. 1 ft. (15) 13 yds. 1 ft. by 8 yds. 2 ft. (14) 6 yds. 1 ft. by 1 yd. 2 ft. (16)

yds. 2 ft. by 7 yds. 1 ft.

Find the area in square feet and inches of rectangles having

the following dimensions:

(17) 2 ft. 6 in. by 3 ft. 8 in.
(18) 13 ft. 6 in. by 8 ft. 7 in.

(19) 11 ft. 2 in. by 3 ft. 10 in. (20) 5 ft. 9 in. by 1 ft. 9 in.

Find the area in square yards, feet, and inches of rectangles having the following dimensions:

(21) 3 yds. 2 ft. 1 in. by 8 yds. 1 ft. 6 in.
(22) 7 yds. 1 ft. 9 in. by 1 yd. 2 ft. 3 in.
(23) 4 yds. 2 ft. 3 in. by 5 yds. 2 ft. 7 in.
(24) 9 yds. 1 ft. 6 in. by 4 yds. 1 ft. 10 in.

Find the area in acres, roods, and poles of rectangles having the following dimensions:(25) 125 lks. by 1080 lks. (26) 1285 lks. by 250 lks. (27) 625 lks. by 175 lks.

(28) 15 ch. 10lks. by 8 ch. 75lks. (29) 7 ch. 19 lks. by 6 ch. 25 lks. (30) 6 ch. 25 lks. by 5 ch. 40 lks.

(31) How many square feet in a floor 13 ft. 8 in. by 15 ft 6 in. ?

(32) Required the acreage of a field 175 links by 225 links. (33) How many acres, roods, and poles are there in a rectangular field whose length is 7 ch. 50 lks., and breadth 8 ch. 30 lks.?

(34) What is the area of a rectangular court of which the length is 250 yds. 1 ft. 6 in., and the width 32 yds. 2 in.?

(35) Required the cost of a mahogany dining-table top, 14 ft. 3 in. long and 4 ft. 8 in. broad, at 1s. 8d. per sq. ft.

(36) A rectangular enclosure is 32 yds. 2 ft. 3 in. long, and 15 yds. 1 ft. 6 in. broad, what will it cost in paving at 6s. per sq. yd.?

(37) Find the side of a square whose area is equal to that of a rectangle 85 ft. 4 in. long, and 5 ft. 4 in. wide.

(38) Two fields, one rectangular and the other square, are measured, and found to contain exactly the same area. The length of the former is 625 links, and its breadth 361 links: what is the length of a side of the latter?

(39) How many acres are there in a field which is a quarter of a mile long and

242 yards wide?
(40) A rectan-
gular grass-plot is
120 ft. long and
80 ft. wide. It is
surrounded by a
gravel path 6 ft.
wide. Find the
area of the path
in square yards.

D

A

H

C

G

B

[See Arts. 16, 17. Find the area of the outer rectangle, ABCD, and also of the inner rectangle, E F G H. Then the difference of these two is the area of the path. To obtain the length of the outer rectangle, twice the width of the path must be added to the length of the inner rectangle. Similarly, twice the width of the path added to the breadth of the inner rectangle, will give the breadth of the outer rectangle.]

(41) Find the expense of paving a road of a uniform breadth of 4 yards round the inside of a rectangular piece of ground, the length of which is 56 yards and breadth 48 yards, the cost of paving a square yard being 1s. 3d.

[In this case, twice the width of the road must be subtracted from the given dimensions.]

(42) If the length of a rectangular field be 880 yds. and its breadth 440 yds., what fraction of a square mile is its area? (43) If a window be 8 ft. 2 in. high and 5 ft. 3 in. wide, how

many panes of glass, each measuring 14 in. by 9 in., will exactly

fill it?

[See Art. 42. Each pane will occupy a space of 14x9=126 sq. in. Find how many times this space is contained in the area of the window.]

(44) A rectangular court 20 yds. 2 ft. 6 in. long by 12 yds. 2 ft. 3 in. broad, is to be paved with flagstones, each of which is 2 ft. 1 in. by 1 ft. 5 in. How many are required?

(45) A railway platform is 54 yards long and 21 yards broad. How many planks does it contain, each being 13 feet long and 10 inches wide?

(46) Find the number of turfs, 4 feet by 8 inches, required for a garden plot 64 feet by 75 feet.

(47) A schoolroom is 60 feet long and 20 feet broad, how many children would it contain, allowing 8 sq. feet of space per child?

(48) A procession is formed of 121 ranks of men, 9 in a rank; if the men were arranged in a solid square, find how many there would be in a side.

(49) A rectangle measures 18 ft. by 24 ft.: find the area of a square which has the same perimeter as the rectangle.

(50) Find the cost of making a road a rod wide and 300 chains long, if the land costs £140 per acre, and the construction £4 4s. per sq. chain.

[1 rod 25 links. 300 chains=30,000 links.]

(51) A rectangle contains 450 sq. ft., and it is twice as long as it is broad; find its sides.

[As the length is twice the breadth, the area of the rectangle will be equal to that of two squares, each containing 225 sq. ft. The sq. root of 225 is 15, therefore the side of each square is 15, and the length and breadth of the rectangle 30 and 15 respectively.]

(52) The area of a rectangular field whose length is three times its breadth, is 45 ac. 3 po.: find its dimensions in links. [As the length is three times the breadth, the field will form three squares, side by side, each having an area of one-third of 45 ac. 3 po.]

(53) The area of a rectangular field contains 975744 sq. ft., and one of the sides is three-and-a-half times as long as the other. What is the length of each side?

(54) A rectangular park, one side of which is twice as long as the other, contains 500 acres. How many sq. yds. of ground will be occupied by a road 15 feet wide running round the outside of the park?

[Reduce the acres to sq. yards, 4810 sq. yds. = 1 acre.]

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