295. A pyramid is a solid formed by several triangular planes united at the same point S, and terminating in the different sides of a plane figure, as ABCDE. The altitude of the pyramid is the line SO, drawn perpendicular to the base. 296. To find the solidity of a pyramid, Multiply the area of the base by the altitude, and divide the product by 3. EXAMPLES. 1. Required the solidity of a pyramid, of which the area of the base is 95 and the altitude 15. 2. What is the solidity of a pyramid, the area of whose base is 260 and the altitude 24? Ans. 2080. 3. What is the solidity of a pyramid, the area of whose base is 207 and altitude 18? Ans. 4. What is the solidity of a pyramid, the area of whose base is 403 and altitude 30? Ans. 4030. 5. What is the solid content of a pyramid, the area of whose base is 270 and altitude 16? Ans. 1440. 295. What is a pyramid ? What is the altitude of a pyramid? 296. How do you find the solidity of a pyramid? 6. A pyramid has a rectangular base, the sides of which are 25 and 12; the altitude of the pyramid is 36: what is its solid content? Ans. 7. A pyramid with a square base, of which each side is 30, has an altitude of 20: what is its solid content? 297. A cone is a round body with a circular base, tapering off to a point called the vertex. The point C is the vertex, and the line CB is called the axis or altitude. Ans. 6000. B 298. To find the solidity of a cone, Multiply the area of the base by the altitude, and divide the product by 3. EXAMPLES. 1. Required the solidity of a cone, the diameter of whose base is 5 and the altitude 10. We first square the diameter and multiply it by .7854 which gives the area of the base. We next multiply by the altitude, and then divide the product by 3. OPERATION. 25 X.7854-19.635 10 3)196.35 Ans. 65.45 2. What is the solidity of a cone, the diameter of whose base is 18 and the altitude 27? Ans. 2290.2264. 3. What is the solid content of a cone, the diameter of whose base is 20 and the altitude 30? Ans. 297. What is a cone? What is the vertex? What is the axis? 298. How do you find the solidity of a cone? 4. What is the solidity of a cone whose altitude is 27 feet, and the diameter of the base 10 feet? Ans. 706.86. 5. What is the solidity of a cone whose altitude is 12 feet, and the diameter of its base 15 feet? Ans. 706.86. GAUGING. 299. The mean diameter of a cask is found by adding to the head diameter, two-thirds of the difference between the bung and head diameters, or if the staves are not much curved, by adding six-tenths. This reduces the cask to a cylinder. Then, to find the solidity, we multiply the square of the mean diameter by the decimal -7854 and the product by the length. This will give the solid content in cubic inches. Then if we divide by 231, we have the content in gallons (ART. 90). Multiply the length by the square of the mean diameter, then by the decimal .7854, and divide by 231. OPERATION. 231 lxd2x-7854 If, then, we divide the decimal .7854 by 231, the quotient carried to four places of decimals is .0034, and this decimal multiplied by the square of the mean diameter and by the length of the cask, will give the content in gallons. 300. Hence, for gauging or measuring casks, we have the following Multiply the length by the square of the mean diameter : then multiply by 34 and point off four decimal places, and the product will then express gallons and the decimals of a gallon. 299. How do you find the mean diameter of a cask? How then do you find the solid content in cubic inches? How in gallons? 300. What is the rule for gauging casks? EXAMPLES. 1. How many gallons in a cask whose bung diameter is 36 inches, head diameter 30 inches, and length 50 inches? We first find the difference of the diameters, of which we take two-thirds and add to the head diameter. We then multiply the square of the mean diameter, the length and 34 together, and point off four decimal places in the product. OPERATION. 36-30-6 of 6 = 4 30+4=34 342=1156 1156 x 50 x 34= 196.52gal. 2. What is the number of gallons in a cask whose bung diameter is 38 inches, head diameter 32 inches, and length 42 inches? Ans. 3. How many gallons in a cask whose length is 36 inches, bung diameter 35 inches, and head diameter 30 inches? Ans. 136 gallons. 4. A water tub holds 147 gallons; the pipe usually brings in 14 gallons in 9 minutes: the tap discharges, at a medium, 40 gallons in 31 minutes. Now, supposing these to be left open, and the water to be turned on at 2 o'clock in the morning; a servant at 5 shuts the tap, and is solicitous to know in what time the tub will be filled in case the water continues to flow. Ans. The tub will be full at 3min. 48114sec. after 6. 5. How many gallons in a cask whose length is 40 inches, head diameter 34 inches, and bung diameter 38 inches? Ans. 1. A merchant bought 13 packages of goods, for which he paid $326: what will 39 packages cost at the same rate? Ans. $978. 2. Two merchants, A and B, traded together; A put in £320 for 5 months, and B £460 for 3 months; they gained £100: how much should each one receive? 49 Ans. A £53 13s. 9135d., B £46 6s. 2114d. 3. If I buy 1000 ells Flemish of linen for £90, what must it be sold for per ell English, to make £10 by the purchase? Ans. 4. If of a gallon of wine cost § of a £, what will of a tun cost? Ans. 5. If an officer's salary is £48 per annum, how much will he receive in 232 days? Ans. £30 10s. 24d.+. 6. If a gentleman spends, one day with another, £1 7s. 10d., and at the end of the year has saved £340, what is his yearly income? Ans. £848 14s. 44d. 7. If 8 cannons expend, in one day, 48 barrels of powder, how much will 24 cannons expend in 22 days? Ans. 8. What number is that which being multiplied by will produce ?. Ans. 9. A person dying divided his property between his widow and his four sons; to his widow he gave $1780, and to each of his sons $1250: he had been 25 years in business, and had cleared on an average $126 a year: how much had he when he began business? Ans. $3567. 10. A besieged garrison consisting of 360 men was provisioned for 6 months, but hearing of no relief at the end of 5 months, dismissed so many of the garrison that the remaining provision lasted 5 months: how many men were sent away? Ans. 288. 11. Two persons, A and B, are indebted to C; A owes $2173, which is the least debt, and the difference of the debts is $371: what is B's debt? Ans. |