PRISMS.

1. A right prism stands upon a triangular base, whose sides are 13, 14, and 15 inches. If the height is 10 inches, find its volume and whole surface. Ans. 840 cu. in.; 4 sq. ft. 12 sq. in.

2. The weight of a brass prism standing on a triangular base is 875 lbs. If the sides of the base are 25 in., 24 in., and 7 in., find the height of the prism, supposing that 1 cu. ft. of brass weighs 8,000 oz. Ans. 3 ft.

3. Water flows at the rate of 30 yards per minute through a wooden pipe whose cross-section is a square on a side of 4 inches. How long will it take to fill a cubical cistern whose internal edge is 6 feet?

Ans. 218 min.

4. Find the volume of a truncated prism (that is the part of a prism included between the base and a section made by a plane inclined to the base and cutting all the lateral edges), whose base is a right triangle, base 3 feet, and altitude 4 feet, and the three lateral edges 3 feet, 4 feet, and 5 feet respectively. [Formula.-V=3A\e1+e2+e3), where A is the area of the base and e, eg, and e, the lateral edges.]

CYLINDERS.

1. How many cubic yards of earth must be removed in constructing a tunnel 100 yards long, whose section is a semi-circle with a radius of 10 feet?

2. Find the convex surface of a cylinder whose height is three times its diameter, and whose volume is 539 cubic inches.

3. The cylinder of a common pump is 6 inches in diameter; what must be the beat of the piston if 8 beats are needed to raise 10 gallons?

Ans. 124 in. 4. A copper wire inches in diameter is evenly wound about a cylinder whose length is 6 inches and diameter 9.9 inches, so as to cover the convex surface. Find the length and weight of the wire, if 1 cu. in. of copper weighs 5.1 oz. Ans. 1,885 in., nearly; 75.5 oz. 5. A cubic inch of gold is drawn into a wire 1,000 yards long. Find the diameter of the wire. Ans. .006 in.

6. The whole surface of a cylindrical tube is 264 square inches; if its length is 5 inches, and its external radius is 4 inches, find its thickness. [Use π=7.]

Ans. 1 in. 7. If the diameter of a well is 7 feet, and the water is 10 feet deep, how many gallons of water are there, reckoning 71⁄2 gallons to the cubic foot?

PYRAMIDS AND CONES.

1. Find the entire surface of a right pyramid, of which the height is 2 feet and the base a square on a side of 1 ft. 8 in. Ans. 10 sq. ft.

2. Find the convex surface of a right pyramid 1 foot high, standing on a rectangular base whose length is 5 feet 10 inches and breadth 10 inches. Ans. 8 sq. ft. 128 sq. in.

3. Find the convex surface of a right pyramid having the same base and height as a cube whose edge is 10 inches. Ans. 223.6 sq. in. 4. Find the weight of a granite pyramid 9 feet high, standing on a square base whose side is 3 feet 4 inches, 1 cubic foot of granite weighing 165 lbs. Ans. 2 tons, 9 cwt. 12 Ïbs.

5. Find the height of a pyramid of which the volume is 623.52 cu. in., and the base a regular hexagon on a side of 1 foot. Ans. 5 inches.

6. The volume of a regular octahedron is 471.41 cubic feet; find the length of each edge. Ans. 10 feet. 7. Find the surface of a regular tetrahedron, if the perpendicular from one vertex to the opposite face is 5 inches.

8. A conical vessel is 5 inches in diameter and 6 inches deep. To what depth will a ball 4 inches in diameter sink in the vessel?

9. The ends of the frustum of a pyramid are squares whose sides are 20 inches and 4 inches, respectively. If its altitude is 15 inches, what is its convex surface? Ans. 110 sq. in.

10. What is the volume of a frustum of a pyramid whose upper base is 4 inches square, lower base 28 inches, and the length of the slant edges 15 inches?

11. The volume of a frustum of a cone is 407 cubic inches and its thickness is 101⁄2 inches? If the diameter of one end is 8 inches, find the diameter of the other end. [T=Y.] Ans. 6 inches.

SPHERES.

1. Find the ratio of the surface of a sphere to the surface: (i) of its circumscribed cylinder, (ii) of its circumscribed cube.

2. A cube and a sphere have equal surfaces; what is the ratio of their volumes? Ans. 72:100, nearly.

3. From a cubical block of rubber the largest possible rubber ball is cut. What decimal of the original solid is cut away?

4. Suppose the earth to be a perfect sphere, 8,000 miles in diameter; to what height would a person have to ascend in a balloon in order to see one-fourth of its surface? [Formula.-h=

1
n

2r
n-2'

where r is the radius of

the earth, and is the part of the earth's surface visible to the observer.

If the part of the earth visible to the observer is, or 1/2, n=2 Р

nch

5. A paring an inch wide is cut from a smooth, round orange an inch and a half in diameter. What is its volume, if it is cut from the orange on a great circle of the orange? Ans. .

6. What would be the volume of a paring cut from the earth on the equator? Ans. Ta3, where a is the width of the paring.

Remark. This is a remarkable fact, since the volume of the paring is independent of the radius of the sphere.

7. If, when a sphere of cork floats in the water, the height of the submerged segment is 3/4 of the radius, show that the weights of equal volumes of cork and water are as 34:44.

Note.-The weight of a floating body is equal to the weight of the liquid it displaces. 8. A vertical cylindrical vessel whose internal diameter is 4 feet, is completely filled with water. If a metal sphere 25 inches in diameter is laid upon the rim of the vessel, find what weight of water will overflow. Ans. 699 lbs., nearly.

9. A conical wine-glass 5 3 inches in diameter and 4 inches deep is filled with water. If a metal sphere 5 inches in diameter is placed in the vessel, what fraction of the whole contents will overflow? Ans..

10. Four equal spheres are tangent to each other. What is the radius of a sphere tangent to each?

11. To what depth will a sphere of ice, three feet in diameter, sink in water, the specific gravity of ice being ?

12. Find the volume removed by boring a 2-inch auger-hole through a

6-inch globe.

13. What is the volume removed by chiseling a hole an inch square through an 8-inch globe?

PRISMATOIDS AND WEDGES.

1. Find the weight of a steel wedge whose base measures 8 inches by 5 inches, and the height of the wedge being 6 inches; if 1 cu. in. of steel weighs 4.53 oz.?

2. Find the volume of a prismatoid of altitude 3.5 cm., the bases being rectangles whose corresponding dimensions are 3 cm. by 2 cm. and 3.5 cm. by 5 cm.

3. The base of a wedge is 4 by 6, the altitude is 5, and the edge, e, is 3. Find the volume.

RINGS.

1. Find the surface and volume of a ring, the radius of the inner circumference being 10%1⁄2 inches and the diameter of the cross-section 31⁄2 inches. Ans. 847 sq. in.; 741 cu. in.

2. Find the surface and volume of a ring, the diameters of the inner and outer circumferences being 9.8 inches and 12.6 inches respectively. Ans. 154.88 sq. in.; 54.21 cu. in.

SIMILAR SOLIDS.

1. The edges of two cubes are as 4:3; find the ratio of their surfaces and their volumes.

2. The surfaces of two spheres are in the ratio of 25:4; find the ratio of their volumes.

3. At what distance from the base must a cone, whose height is 1 foot, be cut by a plane parallel to the base, in order to be divided into two parts of equal volume? Ans. 2.47 in.

4. A right circular cone is intersected by two planes parallel to the base and trisecting the height. Compare the volumes of the three parts into which the cone is divided. Ans. 1:7:19.

EXAMINATION TESTS.

ARITHMETIC.

1. How do you divide one fraction by another? Why is the fraction thus divided?

2. Divide four million and four millionths by one ten-thousandth. Write the answer in figures and words.

3. If a liter of air weighs 1.273 gr., what is the weight in kilos., if the air is in a room which contains 78 cu. m.?

4. The base of a cylinder is 12 inches in diameter and its altitude is 25 inches. Required the solid contents.

5. The edge of a cube is 6 inches; what is the length of the diagonal of the cube?

6. A broker bought stock at 4% discount, and sold it at 5% premium, and gained $450. How many shares did he purchase?

7. A ships 500 tons of cheese, to be sold at 92 cents a lb. He pays his agent 3% for selling; the proceeds are to be invested in sugar, after a commission of 2% is deducted for buying. Required the entire commission.

8. Upon what value are dividends declared? Brokerage estimated? Usual rate of brokerage?

9. What is the face of a note dated July 5, 1881, and payable in 4 months to produce $811, when discounted at 9%?

10. Upon what principle is the United States rule for partial payments based? The Mercantile rule? How does compound interest differ from annual interest? Ohio State List, 1884.

For the benefit of students preparing for county or state examinations, we write out the answers to the above questions as a specimen of how the examination paper ought to be prepared:

SUBJECT: Arithmetic.

Name of Applicant....

(a) Invert the terms of the divisor and then multiply the numerators of the fractions together for the numerator of the quotient and the denominators together for the denominator of the quotient.

(b) The fraction is thus divided because inverting the terms of the divisor gives the number of times the divisor is contained in 1, as is shown by analysis. The number of times then it is contained in any other number is obtained by multiplying this number by the number of times the divisor is contained in 1.

(1. Four million and four millionths=4000000.000004=

40000000000004
1000000

Grade.

40000000000004

100

III.

=400000000000 =400000000000.04=

.. The quotient is four hundred billion and four hundredths.

40000000000004
1000000
4
100

10000

1

5. 1000 g. 1 kilo.

6. 99294 g.=99294 g.÷1000-99.294 kilos.

The weight of 78 cu. m. of air weighs 99.294 kilos.

III.

..

...

III.

3. 1⁄4122=36′′ sq. in., the area of the base of the cylinder. 4. 25X36π=900 cu. in.=900X3.141592×1 cu. in.=

2827.4328 cu. in., the volume of the cylinder. .. The volume of the cylinder is 2827.4328 cu. in.

1.

6 in. the length of the edge of the cube.

2. 36 sq. in.+36 sq. in.=72 sq. in. the area of the square described on the diagonal of one of the equal faces,

which is the sum of the areas of the squares described on two equal edges.

II. 3. 72 sq. in.+36 sq. in.=108 sq. in. area of square described on the diagonal of the cube, which equals the sum of the areas described on the three edges.

III.

4. 6/3 in.=√108×1 in.=10.392+ in., the length of the diagonal of cube.

.. 6/3 in.=10.392+ in.=length of diagonal of cube.

[1. 100% par value of stock.

2.

3.

4.

4% discount.

96%-100%-4%=market value, or cost of stock.
5%-premium.

5. 105%-100% +5%-selling price of stock.

6.

6

II.

7.

7

III.

9%=105%-96%=gain.

$450=gain.

8. .9% $450.

9.

1%=$50, and

10. 100% $5000=par value of stock.

11. $100 par value of one share, usually. Then | 12. $5000=par value of $5000÷$100, or 5 shares. .. He purchased 5 shares.

(1. 91⁄2 cents=selling price of one lb.

2. $47.50=500×$0.09%1⁄2=selling price of one lb.
100% $17.50.

$47.50-$0 95-$46.55-proceeds, or the amount to be invested in sugar.

2%=2X$0.45 $0.90 commission on sugar.

10. .. $0.95+$0.90 $1.85 total commission.
.. $1.85 entire commission.