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G's tax may now be obtained from the table, where it is found, that $40 pays $8 8
Value ofG's inventory, $48,67 “ $9,734 G's tax, - <2000sMEASURING SURFACES AND SOLIDs.
To find the superficial contents of surfaces which are - not rectangles.
1. How many square feet in a board 18 feet long, 20 inches wide at one end, and 10 at the other?
Explanation.—It is evident from plate, page 272,
that the average width of the board will be the width of the middle. It may also be proved by the same plate, that half the sum of the ends will be the width in the middle. Ans. 22}.
2. How many square feet in six boards, each 20 feet long, 15 inches wide, at one end, 3 inches at
the other ? Ans. 90. 3. How much land in a field 49 rods long, 35 rods wide at one end, 25 at the other, if the angles at the extremities of the first mentioned side, be right ones, and all the sides be right lines? - Ans. 9 A.30 rods.
It has already been shown, that multiplying the length of a rectangle by the breadth, gives the area. The surface of a right angled triangle is equal to one half the surface of a rectangle, as long as the base of the triangle, and as wide as the perpendicular; therefore multiplying the base of any surface in the form of a right angled triangle by half the length of the perpendicular, gives the area.
4. How much land in a field in the form of a right angled triangle, if the base be 49% rods, and the perpendicular 23 rods 7
When parts of a rod are given in a vulgar fraction, or feet, it is the better way to change them to a decimal fraction. Ans. 3 A. 2 R. 94 rods.
5. How many feet of boards will be required to board the gable ends of a building, 40 feet wide, if a perpendicular from the beam to the ridge be 9 feet, 7 inches 2 Ans. 3834.
JNote—The contents of any irregular surface bounded by right lines, may be found by dividing it into right angled triangles, and adding the contents of the triangles so formed.
To find the contents of circles.
Multiply half the circumference by half the diameter.
6. If the diameter of a circle be 8 feet, what is the area 7 Ans. 50+ feet. 7. How many square feet on the surface of the end of a cylinder, whose circuinference is 10 feet 2 - Ans. 7##. 8. How many rods of land will it require to make a walk one rod in width, round a circular pond, whose diameter is 10 rods 7 Ans. 344.
To find the contents of timber, or any other solid body, when the same bigness is preserved throughout the length of the same.
Multiply the number of square inches, feet, &c. on the surface of one end, by the length.
This rule is founded on the principle illustrated, page 97.
9. How many solid feet in a stick of timber 25 feet long, 24 feet wide, and 1 ft. 9 in. thick? Ans. 109,375. 10. How many solid feet in a round stick of timber, 164 feet long, and 24 in diameter 2 Ans. 81+3+. 11. What will a round stick of timber come to, at $5 per ton, if it be 20 feet long, and 7 feet in circumference 2 Ans. $9,73+. 12. How many solid feet in a stone, 14 feet long, having the ends in the form of a right angled triangle, the widest side being 15 inches, and one of the others 12 2 Also, what will the hewing come to at 44 cents per square foot? Ans. 5; solid feet; and cost of hewing $1,924-. 13. H employed T to dig and stone a well, 30 feet deep, and 9 feet diameter; and was to give him 4 cents for each solid foot of stone, it should require in stoning the same. The quantity of stone was estimated at # of the whole space except the tunnel, which was 3} feet diameter. How much did T receive 7 Ans. $48,6144.
To find the contents of solids larger at one end than the other, and when the size regularly decreases,
If the stick decrease with a true slant till it terminates in a point, multiply the surface of the base, or larger end, by one third of the length. . But if the body do not taper to a point multiply the surface of the larger end by the length the body would be if it continued to a point, and subtract the contents of the supposed part from the whole contents. The contents of the supposed part is found by multiplying the surface of the smaller end of the given part, into one-third of the length of the supposed part. JNote When the body does not taper to a point, the length it would be if it did, may be found by the following proportion. As the difference between the smaller diameter, or thickness, and the larger diameter, or thickness, is to the length of the given stick, so is the diameter, or thickness, of the larger end to the length required. The given length taken from the whole, will leave the length of the supposed part. 14. What is the contents of a round stick of timber, 15 feet long, the surface of one end being 24 feet, and the other a point % Ans. 124 feet. 15. How many solid feet in a stick of timber, whose base is 20 inches square, and its length 60 feet, if it decrease to a point Ans. 55#. 16. How much timber in a stick, 23 feet long if the larger end be two feet square, and the other 1,5 feet 7 Ans. 1 T. 2044 feet.
.4 more concise method of finding the contents of a frustum of a cone or pyramid, that is, of bodies tapering with a true slant, when they do not ... to a point, may be found in the following 3rotes
Add together the areas of each end and their
mean proportional - the o: ional, and multiply their sum by , of }
17. Required the contents of a round stick of timber, 174 ft. long, the diameter of the larger end being 24 feet, and that of the smaller end, 1 foot ? Ans. 44}} feet. 18. If a stick of timber, 60 feet long, and 14 feet square at the large end, tapering to a point at the other, be sawed off in the middle, how many more solid feet will there be in the larger part than in the smaller. Ans. 33,75. 19. A gentleman purchased two stone pillars for 3,25 dollars per solid foot. The base of each was 2 feet in diameter, the other end 14 feet, and the length 12 feet. What was the cost of both pillars? - Ans. $164,645-H.
The value ofthe pound, shilling, pence, and farthing, varies in different parts of the United States. The value is determined by the number of shillings in a dollar, as the value of a dollar is the same in all the States.
The following table exhibits the number of shillings in each state, of which the dollar is composed.
Dollar. In New-England, - - - - “ Virginia, - - - - - “ Kentucky, - - - - - “Tennessee, - - - - - “ New-York, - - - - - “ North-Carolina, - - -