Observe-To get four places exact it is well to carry on the extraction of the root to six places. Ex. 2. The diameter of a circle is 650 cm., what is its area? 11 sq. cm. 14 (cm. diam.)2, 650 cm. diam. i.e., i.e., = Ex. 3. How much will it cost to cover a circular plot of ground 130 feet in circumference with gravel at 4d. per square yard? Because we have (foot rad.)2 the square of the preceding multipliers must be taken, hence (65)2 × 4 we find that the value is very nearly 600 pence, Ex. 4. Calculate the area of that zone of the earth's surface which lies between latitude 30° and latitude 45°, assuming the earth's radius to be 4,000 miles, and π to be 3.1416. 1 2 × 3·1416 × 4,000 × 4,000—2—3) sq. miles, Answer--2.082 × 107 square miles. Ex. 5. With reference to a certain map it is known that a inch represents 2844 acres. What is the scale of the map? 2,560 acres 9 inch2, square .. i.e., i.e., = EXERCISE XII. 1. Find the area in acres of an isosceles right-angled triangle, the perpendicular from the right angle on the hypotenuse being 1,000 feet. 2. Find in hectares the area of a square of which the diagonal is 10 chains, assuming that a square metre is 1·196 of a square yard. 3. A rectangular enclosure is one acre in extent, and its perimeter is 322 yards. Find the lengths of its sides. 4. The diagonal of a square is 20 chains. Required the area in acres. 5. Find the area of a field which has the form of an equilateral triangle, and has one side 350 yards in length. 6. Calculate the area of the triangle whose sides are 11, 60, and 61 feet. 7. The two diagonals of a quadrilateral are 30 and 40 chains, and the angle between them is 30°. Required the area in acres. 8. A field is in the form of a trapezium, the two parallel sides being 6 chains and 4 chains; a third side is 5 chains, and is inclined to a parallel side at 60°. Calculate the area in acres. 9. A square bowling-green, 50 yards in the side, has a uniform slope round it, the slant height of which is 2 yards, and the angle which the slope makes with a horizontal plane is 30°. Find the cost of turfing the green and slope when the work is done at the rate of d. per square yard. 10. Find to the nearest square foot the number of square feet of lead required to cover a pyramidal roof, the base being a square of 19 feet in the side, and the height 6 feet. 11. The diameter of a circle is 135 mm. What is its circumference, and its area? 12. The circumference of a circle is 600'04 cm. area? What is its diameter and its 13. The area of a circle is 55,155 ares. What is its diameter and its circumference? 14. Express 36 circular inches in terms of square inches; and 63 square inches in terms of circular inches. 15. A penny and a halfpenny have diameters of one-tenth of a foot and of an inch respectively. If a halfpenny lie wholly on the top of a penny, what amount of the upper surface of the penny will be left uncovered? 16. An elliptic plot is described in a garden by means of a string 20 feet in length, and passing round two pegs distant by 5 feet. What is the area of the plot? 17. Calculate the area of an elliptical pond, the major axis of which is 250 feet, and the minor axis 150 feet. 18. Find the area of an ellipse whose axes are 25 chains and 22 chains 25 links. 19. Berlin is 11° 35′ east of Paris. Find the area of the portion of the earth's surface between their meridians; the earth's radius being taken as 3,963 miles. 20. On a certain map a square inch represents 4,000 acres. What is the linear scale? 21. The vertical scale of a drawing is 40 feet to an inch, and the horizontal 400 feet to an inch. What is the scale for the area? 22. On a map it is found that 100 acres are represented by 359 48 square inches; but the scale not being attached, it is required to calculate what it is. Give the scale in the form of a ratio, and also in terms of foot per mile. SECTION XIII.---VOLUME. ART. 90.-General Unit of Volume. Any unit of volume may be denoted by V. The systematic unit is the volume of a cube whose side is the unit of length, and in such case we have 1 V = L3, where L3 denotes the same as cubic L. When the unit is not systematic we have some other number instead of 1. ART. 91.-Imperial Units of Volume. In the imperial system we have cubic units and units of capacity. The base of the cubic unit is the cubic yard, and the relations to it of the cubic foot and the cubic inch follow from the relations of the linear foot and the linear inch to the linear yard. The primary unit of capacity is the gallon, which involves in its definition the standard of mass. It is the volume of ten imperial pounds of distilled water at the temperature of 62° Fahr. Further, the mass of the water is to be determined by weighing in air against brass weights, the air also being at the temperature of 62° Fahr., and the barometer standing at 30 inches. In the original definition of the gallon, the volume defined as above was stated to be equivalent to 277-274 cubic inches; but when a more accurate determination of the density of water was made, the alternative part of the definition was repealed. According to the most recent determinations1 the gallon is equivalent to 277.123 cubic inches. The brass gallon, marked "imperial 1 Rankine's Rules and Tables, p. 99. standard gallon," constructed when the gallon was originally defined, is not the ultimate standard, but pure water taken in conjunction with the standard of mass. The other units of capacity are defined by means of the gallon. ART. 92. Metric Units of Volume. In the metric system we have three series of units of volume. The stere and its derivatives are for solid measure, as for example the measuring of wood; the litre and its derivatives are for fluid measure or measure of capacity; while the cubic series is the best adapted for calculations and for science generally. The stere is by definition equivalent to a cubic metre, and the litre to a cubic decimetre. Their derivatives are decimal; while those of the cubic series are millesimal. The authorized abbreviation for cubic is the index3, as in cm.3 for cubic centimetre. That for stere is s., and for litre l. In the C.G.S. system the primary unit of volume is the cubic centimetre. |