Ex. 3. The area of a rectangle is 2,592 square feet. If the breadth were increased and the length diminished by 3 feet, we should have a square. Find the length and breadth. which reduces to the quadratic equation hence x2+6x-2592 = 0; x= −3±/9+ 2592; x = 48 is the possible solution. Answer-48 feet broad, 54 feet long. Ex. 4. A farm in the form of a square contains 400 acres ; what is the length of a side expressed in terms of the mile? 1 mile2 = 640 acres, Ex. 5. Find the cost of papering the walls of a room 22 feet 6 inches long, 18 feet 9 inches wide, and 11 feet high; 99 square feet not requiring to be papered; the paper being 2 feet 9 inches wide, and costing 3d. per linear yard. For a side = 1 square foot foot long by foot high, 22.5 ft. long by 11 ft. high; 22.5 × 11 square foot. Hence for the two sides 2 × 22.5 × 11 square feet, Hence 2 × 11 (22·5 + 18·75) − 99 square feet of paper, 1 penny = 1 foot long ; pence, EXERCISE XI. 1. The areas of England, Scotland, Ireland, are 58,311, 30,463, 31,754 square miles respectively; express these areas in terms of the square kilometre. 2. A square, whose side is 146 feet, contains 1980-25 square metres. Find the number of inches in a metre to three decimal places. 3. Give that a metre equals 3·2809 ft.; find how many square metres there are in 1,000 square yards. 4. How many persons could be seated in an area ten miles square, each person occupying half a square metre? 5. What is 21. per acre in terms of franc per hectare, when exchange is at 25 fr. 20 cts. per l.? 6. A plot of land is sold at 1,2001. per acre. square metre, taking 1l. = 25 francs? What is the price in franos per 7. Seventy-five per cent. of the area of a farm is arable; of the remainder, eighty-five per cent. is pasture, and the rest is waste; the area of the waste is 3 acres 0 roods 20 poles. What is the area of the farm? The one is one third of a mile 8. Find the number of granite blocks required to pave a street which is 1 mile long and 16 yards wide, the block being 4 inches broad and 12 inches long. 9. Two rectangular fields are equal in area. long by 990 feet wide; the other is exactly square. 10. The sides of a rectangular piece of land are in the ratio of 2 to 7. What is the length of each side if the area contains 341,373,816 square feet? 11. The length of a rectangle is to its breadth as 4 to 3. If we increase the length by 3 feet and the breadth by 4, we increase the area by 287 feet. Required its dimensions. 12. The length of a rectangular piece of ground is twice its breadth; its area is 30,479 805 square feet. Find its length and breadth. 13. Find the length of the side of a square which contains 1 acre. 14. Calculate to the nearest integer the number of feet in each side of a square field 4 statute acres in extent. 15. If 640 acres go to a square mile, what is the length of each side of a square piece of land which contains 100 acres? 16. Find the length of the side of a square field which contains 7 acres 3 roods 15 perches. 17. A room 21 feet long required 49 yards of carpet of a yard broad. Find the breadth of the room.. 18. The number of yards of paper required to cover the four walls of a room 54 feet wide and 30 feet high is 880, and the breadth of the paper is of a yard. Required the length of the room. 19. The first of two pictures is 1 ft. 6 in. by 2 ft., the second 2 ft. by 2 ft. 6 in., and they are to be framed in the same way; if the glass and frame of the former cost 78. 6d. and that of the latter 11s. 2d., what is the price of the glass per square foot, and of the frame per foot of length? 20. A merchant buys cotton (27 inches in width) at 5 cents per square yard. He pays a duty of 2 cents per square yard, and 15 per cent. ad valorem. For what price per yard should he sell it in order to gain 25 per cent. on his outlay? 21. Of two squares of carpet, one measures 44 feet more round than the other, and 187 square feet in area. What are their sizes? 22. A square field of grain containing ten acres is to be cut down by a reaper working round and round; the cut of the reaper is 6 feet. How many rounds must the reaper take before the field is half cut? 23. A square plot of ground, 21 yards in the side, is sold for the greatest number of sovereigns which can be placed flat upon it. Find the price, the diameter of the sovereign being seven eighths of an inch. 24. What length of fence is required to enclose 100 acres; first, when the land is in the form of a square; second, when in the form of a rectangle, having length 23 times breadth? B SECTION XII-SURFACE, Continued. ART. 82.-Area of the Sector of a Circle. Fig.7. Consider any sector of a circle ABC; draw the chord BC (Fig. 7). The area of the triangle ABC is a first approximation to the area of the sector ABC. Bisect the sector by the line AD; join BD, DC; the sum C of the areas of the triangles ABD and ADC is a second approximation to the area of the sector. Bisect each of these again; the sum of the areas of the four triangles will be a third approximation to the area of the sector. When the bisection has been continued a large number of times, the altitude of each of the triangles will not differ sensibly from the radius of the sector, and the sum of the lengths of the bases will not differ sensibly from the arc of the sector. Hence the then, by multiplying these two equivalences together, L arc ap pears common to both sides, and may therefore be eliminated, and we obtain S = (L radius)2. ART. 83.—Area of a Circle. For a complete circle becomes 27; hence for a circle ART. 84.—Area of an Ellipse. The circle described with the ajor axis AA' for diameter is called the auxiliary circle (Fig. 8). he area of that circle is given by he breadth of the ellipse is everywhere, as at PM, derived from e corresponding breadth of the circle, as QM, according to the tio by which the minor axis BB' is derived from the diameter of the circle DD'. Hence the area of the ellipse itself is given by or TSL semi-major axis by L semi-minor axis. ART. 85.-Surface of the Common Cylinder. The flat portion of the surface consists of two equal and parallel circles (Fig. 9). If the curved surface were cut parallel to the axis, unrolled, and flattened out, it would form a rectangle having the circumference of the cylinder for length and the axis for breadth. Hence the area of the curved surface is given by 1 SL circumference by L axis. The dependence of the circumference on the radius of the cylinder is given by 2 S by L circumf. = L circumf. by L axis by L radius, i.e., 2 SL axis by L radius. ART. 86. Surface of the Common Cone. The flat portion of the surface is a circle (Fig. 10). If the curved surface were cut along the slant height, unrolled, and flattened out, it would form a sector of a circle, having the slant height for radius and the circumference of the base for arc. Hence (Art. 82) the area of its curved surface is given by |