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157. What is the area of a triangle, of which the base is 30 rods, and the perpendicular 10 rods? Ans. 150 rods.

158. If the area be 150 rods, and the base 30 rods, what is the perpendicular?

Ans. 10 rods.

159. If the perpendicular be 10 rods, and the area 150 rods, what is the base?

Ans. 30 rods.

When the legs (the base and perpendicular) of a rightangled triangle are given, how do you find its area?

When the area and one of the legs are given, how do you find the other leg?

Note. Any triangle may be divided into two right-angled triangles, by drawing a perpendicular from one corner to the opposite side, as may be seen by the annexed figure.

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Here ABC is a triangle, divided into two right-angled triangles, Ad C, and dB C; therefore the whole base, A B, multiplied by one half the perpendicular d C, will give the area of the whole. If AB = 60 feet, and Ans. 480 feet

C= 16 feet, what is the area ?

160. There is a triangle, each side of which is 10 feet what is the length of a perpendicular from one angle to its opposite side? and what is the area of the triangle ?

Note. It is plain, the perpendicular will divide the opposite side into two equal parts. See T 109.

Ans. Perpendicular, 8'66 + feet; area, 43'3 + feet. 161. What is the solid contents of a cube measuring 6 feet on each side? Ans. 216 feet.

When one side of a cube is given, how do you find its solid contents?

When the solid contents of a cube are given, how do you find one side of it?

162. How many cubic inches in a brick which is 8 inches long, 4 inches wide, and 2 inches thick? in 2 bricks? in 10 bricks? Ans. to last, 640 cubic inches. 163. How many bricks in a cubic foot? in 40 cubic Ans. to last, 27000.

eet? in 1000 cubic feet?

164. How many bricks will it take to build a wall 40 feet m length, 12 feet high, and 2 feet thick? Ans. 25920

165. If a wall be 150 bricks, = 100 feet, in length, an 4 bricks, = 16 inches, in thickness, how many bricks wi lay one course? 2 courses ?- 10 courses? If the

Y

wall be 48 courses, = 8 feet, high, how many bricks will build it? 150 × 4 = 600, and 600 × 48 = 28800, Ans.

166. The river Po is 1000 feet broad, and 10 feet deep, and it runs at the rate of 4 miles an hour; in what time will it discharge a cubic mile of water (reckoning 5000 feet to the mile) into the sea? Ans. 26 days, 1 hour.

167. If the country, which supplies the river Po with water, be 380 miles long, and 120 broad, and the whole land upon the surface of the earth be 62,700,000 square miles, and if the quantity of water discharged by the rivers into the sea be every where proportional to the extent of land by which the rivers are supplied; how many times greater than the Po will the whole amount of the rivers be? Ans. 1375 times.

168. Upon the same supposition, what quantity of water, altogether, will be discharged by all the rivers into the sea in a year or 365 days? Ans. 19272 cubic miles.

169. If the proportion of the sea on the surface of the earth to that of land be as 101 to 5, and the mean depth of the sea be a quarter of a mile; how many years would it take, if the ocean were empty, to fill it by the rivers running at the present rate ? Ans. 1708 years, 17 days, 12 hours.

170. If a cubic foot of water weigh 1000 oz. avoirdupois, and the weight of mercury be 134 times greater than of water, and the height of the mercury in the barometer (the weight of which is equal to the weight of a column of air on the same base, extending to the top of the atmosphere) be 30 inches; what will be the weight of the air upon a square foot? a square mile? and what will be the whole weight of the atmosphere, supposing the size of the earth as in questions 166 and 168?

Ans.

2109'375 lbs. weight on a square foot.

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mile.

of the whole atmosphere.

171. If a circle be 14 feet in diameter, what is its circumference?

Note. It is found by calculation, that the circumference of a circle measures about 34 times as much as its diameter, or, more accurately, in decimals, 3'14159 times. Ans. 44 feet. 172. If a wheel measure 4 feet across from side to side, how many feet around it ?

Ans. 124 feet. Ans. 462 feet

173. If the diameter of a circular pond be 147 feet, what is its circumference ?

174. What is the diameter of a circle, whose circumference is 462 feet? Ans. 147 feet. 175. If the distance through the centre of the earth, from side to side, be 7911 mites, how many miles around it?

7911 × 3614159 = 24853 square miles, nearly, Ans. 176. What is the area or contents of a circle, whose diameter is 7 feet, and its circumference 22 feet ?

Note. The area of a circle may be found by multiplying the diameter into the circumference. Ans. 38 square feet. 177. What is the area of a circle, whose circumference is 176 rods? Ans. 2464 rods.

178. If a circle is drawn within a square, containing 1 square rod, what is the area of that circle?

Note. The diameter of the circle being 1 rod, the circumference will be 3'14159. Ans. '7854 of a square rod, nearly. Hence, if we square the diameter of any circle, and multiply the square by '7854, the product will be the area of the circle.

179. What is the area of a circle whose diameter is 10 rods? 102 × 7854 = 73654. Ans. 78'54 rods. 180. How many square inches of leather will cover a ball 32 inches in diameter ?

Note. The area of a globe or ball is 4 times as much as the area of a circle of the same diameter, and may be found, therefore, by multiplying the whole circumference into the whole diameter. Ans. 384 square inches.

181. What is the number of square miles on the surface of the earth, supposing its diameter 7911 miles?

7911 × 24853 = 196,612,083, Ans. 182. How many solid inches in a ball 7 inches in diameter? Note. The solid contents of a globe are found by multiplying its area by part of its diameter. Ans. 1793 solid inches. 183. What is the number of cubic miles in the earth, supposing its diameter as above?

Ans. 259,233,031,435 miles.

184. What is the capacity, in cubic inches, of a hollow globe 20 inches in diameter, and how much wine will it contain, 1 gallon being 231 cubic inches?

Ans. 4188'8 + cubic inches, and 18'13 + gallons. 185. There is a round log, all the way of a bigness; the areas of the circular ends of it are each 3 square feet;

how many solid feet does 1 foot in length of this log contain ? -2 feet in length 3 feet ? 10 feet? A solid of this form is called a Cylinder. How do you find the solid content of a cylinder, when the area of one end, and the length are given ?

186. What is the solid content of a round stick, 20 feet long and 7 inches through that is, the ends being 7 inches in diameter ?

Find the area of one end, as before taught, and multiply it by the length.

Ans. 5'347+ cubic feet.

If you multiply square inches by inches in length, what parts of a foot will the product be ? if square inches by feet in length, what part ?

187. A bushel measure is 18'5 inches in diameter, and 8 inches deep; how many cubic inches does it contain ? Απε. 21504+. It is plain, from the above, that the solid content of all bodies, which are of uniform bigness throughout, whatever may be the form of the ends, is found by multiplying the area of one end into its height or length.

Solids which decrease gradually from the base till they come to a point, are generally called Pyramids. If the base be a square, it is called a square pyramid; if a triangle, a triangular pyramid; if a circle, a circular pyramid, or a cone The point at the top of a pyramid is called the vertex, and a line, drawn from the vertes perpendicular to the base, is called the perpendicular height of the pyramid.

The solid content of any pyramid may be found by multiplying the area of the base by of the perpendicular height.

188. What is the solid content of a pyramid whose base is 4 feet square, and the Ans. 18 feet.

perpendicular height 9 feet ?

4* X = 48.

189. There is a cone, whose height is 27 feet, and whose base is 7 feet in diameter; what is its content?

Ans. 346 feet.

how

190. There is a cask, whose head diameter is 25 inches, bung diameter 31 inches, and whose length is 36 inches; how many wine gallons does it contain ? many beer gallons ?

Note. The mean diameter of the cask may be found by adding 2 thirds, or, if the staves be but little curving, 6 tenths, of the difference between the head and bung diameters, to the head diameter. The cask will then be reduced to a cylinder.

Now, if the square of the mean diameter be multiplied by '7854, (ex. 177,) the pro duct will be the area of one end, and that, multiplied by the length, in inches, will give the solid content, in cubic inches, (ex. 185,) which, divided by 231, (note to table, wine meas.) will give the content in wine gallons, and, divided by 282, (note to table, beer ineas.) will give the content in ale or beer gallons.

In this process we see, that the square of the inean diameter will be multiplied by '7854, and divided, for wine gallons, by 231. Hence we may contract the operation by only multiplying their quotient (7854 231 = '0034;) that is, by '0034, (or by 34, pointing off 4 figures from the product for decimals.) For the same reason we may, for beer gallons, multiply by (7854 = '0028, nearly,) '0028, &c.

282

Hence this concise RULE, for gauging or measuring casks, -Multiply the square of the mean diameter by the length; multiply this product by 34 for wine, or by 28 for beer, and, pointing of four decimals, the product will be the content in gallons and decimals of a gallon.

In the above example, the bung diameter, 31 in. - 25 in. the head diameter = 6 in. difference, and of 64 inches; 25 in. +4 in. 29 in. mean diameter.

Then, 292841, and 841 × 36 in. = 30276.
Then, 30276×24 = 1029384.
30276×28847728.

Ans. 102-9384 wine gallons. Ans. 84-7728 beer gallons. 191. How many wine gallons in a cask whose bung diameter is 36 inches, head diameter 27 inches, and length 45 inches? Ans. 166-617.

192. There is a lever 10 feet long, and the fulcrum, or prop, on which it turns, is 2 feet from one end; how many pounds weight at the end 2 feet from the prop, will be balanced by a power of 42 pounds at the other end, 8 feet from the prop

Note. In turning around the prop, the end of the lever 8 feet from the prop will evidently pass over a space of 8 inches, while the end 2 feet from the prop passes over a space of 2 inches. Now, it is a fundamental principle in mechanics, that the weight and power will exactly balance each other, when they are inversely as the spaces they pass over. Hence, in this example. 2 pounds. 8 feet from the prop, will balance & pounds 2 feet from the prop; therefore, if we divide the distance of the POWER from the prop by the distance of the WEIGHT from the prop, the quotient

pounds ?

will always express the ratio of the WEIGHT to the POWER; 2 = 4, that is, the weight will be 4 times as much as the power. 42X4168. Ans. 168 pounds. 193. Supposing the lever as above, what power would it require to raise 1000 Ans. 1900-250 pounds. 194. If the weight to be raised be 8 times as much as the power to be applied, and the distance of the weight from the prop be 4 feet, how far from the prop must the Ans. 20 feet 195. If the greater distance be 40 feet, and the less of a foot, and the power 175 Ans. 14000 pounds. 196. Two men carry a kettle, weighing 200 pounds; the kettle is suspended on a pole, the bale being 2 feet 6 inches from the hands of one, and 3 feet 4 inches from the hands of the other; how many pounds does each bear ?

power be applied 1

pounds, what is the weight ?

Ans.

1144 pounds. 855 pounds.

197. There is a windlass, the wheel of which is 60 inches in diameter, and the axis, around which the rope coils, is 6 inches in diameter; how many pounds on the axle will be balanced by 240 pounds at the wheel?

Note. The spaces passed over are evidently as the diameters, or the circumfer. ences; therefore, 660 = 10, ratio. Ans. 2400 pounds. 198. If the diameter of the wheel be 60 inches, what must be the diameter of the axle, that the ratio of the weight to the power may be 10 to 13 Ans. 6 inches. Note. This caiculation is on the supposition, that there is no friction, for which it is usual to add to the power which is to work the machine.

199. There is a screw, whose threads are I inch asunder, which is turned by a lever 5 feet inches, long; what is the ratio of the weight to the power ?

Note. The power applied at the end of the lever will describe the circumference of arcle 60 × 2 = 120 inches in diameter, while the weight is raised 1 inch: therefore, the ratio will be found by dividing the circumference of a circle, whose diameter is twice the length of the lever, by the distance between the threads of the

screw.

120 × 3+ = 377+ circumference, and

1

3777
= 3774, ratio, Ans.

1

200. There is a screw, whose threads are of an inch asunder; if it be turned by a ever 10 feet long, what weight will be balanced by 120 pounds power? Ans. 362057 pounds. 201. There is a machine, in which the power moves over 10 feet, while the weight is raised 1 inch; what is the power of that machine, that is, what is the ratio of the Ans. 120.

weight to the power ? 202. A man put 20 apples into a wine gallon measure, which was afterwards filled by pouring in I quart of water; required the contents of the apples in cubic inches.

Ans. 1734 inches. 203. A rough stone was put into a vessel, whose capacity was 14 wine quarts, which was afterwards filled with 24 quarts of water; what was the cubic content of the stone ? Ans. 664 inches.

FORMS OF NOTES, BONDS, RECEIPTS, AND

ORDERS.

NOTES.

No. I.

Overacan, Sept. 17, 1802.

For value received. I promise to pay to OLIVER BOUNTIFUL or order sixty-three dollars fifty-four cents. on demand, with interest after three maths. Attest, TIMOTHY TESTIMONY. WILLIAM TRUSTY.

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