22. Circumference of upper base Circumference of lower base One-half sum Multiplying by slant height gives And the area of the lower base, Total in square yards, or ([(3+5) × 6]+9+25) × 3.1416. The slant height of the whole cone 27 ft. 9 ft. =36 ft. 24. The convex surface of the whole cone = (8×3.1416 ×36) sq. ft.; of the part cut off= (6×3.1416×27) sq. ft. 1287. A sphere (a croquet ball, for instance) and a hemisphere should be used to illustrate these problems. On the plane face of the latter can be drawn the lines AD, FG, HI, CI, etc.; while on the curved face can be drawn HYI, FXG, etc. 25. of 25,000 miles. 26. IH chord of 60° of the great circle radius of the great circle = 4000 miles. IB = 1⁄2 of IH = 2000 miles. 27. The diameter, HI, of the small circle is diameter FG of the great circle; the circumference HYI of 25,000 miles, or 12,500 miles. = 28. The length of a degree of longitude on the 60th parallel is about one-half of the length of a degree on the equator. (See Art. 995, Problem 10.) 29. On the plane face of the hemisphere suggested above (Art. 1287), draw diameters AD and FG at right angles; and 45° from G, a chord NM parallel to FG. (This chord will not bisect AC) As MCG W 45. M 45 1289. The pupils have already learned that the volume of a rectangular prism is equal to the area of the base multiplied by its altitude; these problems are intended to show that the same is true of all prisms and of the cylinder (6). 1292. 8. The volume of the frustum is obtained by deducting the volume of the part cut off from the volume of the whole pyramid (Problem 7). The rule is given later. 10. Fig. 49 gives the method of calculating the slant height. The illustration shows a sec 5 in. B 5 in. from A will fall on CD at a FIG. 49. point X, 5 in. from C. In the right-angled triangle AXC, AX= altitude = 12 in. AC = AX+XC2=144 + 25 = 169; AC= √169= 13. SUPPLEMENT DEFINITIONS, PRINCIPLES, AND RULES A Unit is a single thing. A Number is a unit or a collection of units. The Unit of a Number is one of that number. Like Numbers are those that express units of the same kind. An Abstract Number is one in which the unit is not named. Arabic Notation is expressing numbers by figures. A figure standing alone, or in the first place at the right of other figures, expresses ones, or units of the first order. A figure in the second place expresses tens, or units of the second order. A figure in the third place expresses hundreds, or units of the third order; and so on. A Period is a group of three orders of units, counting from right to left. RULE FOR Notation. Begin at the left, and write the hundreds, tens, and units of each period in succession, filling vacant places and periods with ciphers. RULE FOR NUMERATION. - Beginning at the right, separate the number into periods. Beginning at the left, read the numbers in each period, giving the name of each period except the last. ADDITION Addition is finding a number equal to two or more given numbers. Addends are the numbers added. The Sum, or Amount, is the number obtained by addition. PRINCIPLE. can be added. RULE. Only like numbers, and units of the same order Write the numbers so that units of the same order shall le in the same column. Beginning at the right, add each column separately, and write the sum, if less than ten, under the column added. When the sum of any column exceeds nine, write the units only, and add the ten or tens to the next column. Write the entire sum of the last column. SUBTRACTION Subtraction is finding the difference between two numbers. The Minuend is the number from which the subtrahend is taken. The Remainder, or Difference, is the number left after subtracting one number from another. PRINCIPLES. can be subtracted. · Only like numbers and units of the same order The sum of the difference and the subtrahend must equal the minuend. RULES.-I. Write the subtrahend under the minuend, placing units of the same order in the same column. |