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To find the curve surface of the frustum of a cone.

Multiply the sum of the diameters of the two ends by the slant side; then multiply by 11 and divide by 7.

Ex.

Find the curve surface of the frustum of a cone whose top and bottom diameters are 12 ft. 2 in., and 15 ft. 4 in., and the slant side 17 ft. 6 in.

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EXAMPLES.

I. Find the curve surface of a cone whose diameter is 13 ft. 4 in., and slant side 18 ft. 8 in.

Ans. 3914 feet.

2. Find the curve surface of a cone whose diameter is 19 ft. 2 in., and altitude 21 feet.

Ans. 695 feet, nearly.

3. Find the curve surface of a cone whose altitude is 1 yards, and diameter yard; and the cost of gilding the same at d. per square inch, including the base.

Ans. Surface 163 feet, cost, £3. Is. 1d.

4. The sides of a right-angled triangle are 3, 4, 5; if it revolve about the hypothenuse, find the whole surface generated by the other two sides. Ans. 524 inches.

5. Find the curve surface of the frustum of a cone whose top and bottom diameters are 5 ft. 4 in. and 8 ft. 3 in., and the slant side 10 ft. 6 in. Ans. 224 feet.

6. Find the curve surface of the frustum of a cone whose top and bottom diameters are 15 ft. 4 in., and 26 ft. 8 in., and the depth 23 ft. 9 in. Ans. 1611 feet.

XXXII.

To find the solidity of a cone.

Multiply the square of the diameter by the height; then multiply by 11 and divide by 42.

Ex. Find the solidity of a cone whose diameter is 7 ft. 6 in., and height 14 ft. 7 in.

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To find the solidity of the frustum of a cone.

Add the squares of the diameters of the two ends and their product together; multiply the sum by the height of the frustum; then multiply by II and divide by 42.

Ex. Find the solidity of the frustum of a cone whose top and bottom diameters are 9 and 16 inches, and the depth 14 inches.

92 = 81

162 = 256

9 × 16 = 144

481

14

6734

6734

6 74074

7 123453

1763 Ans.

EXAMPLES.

I. Find the solidity of a cone whose diameter is 10 yards, and altitude 11 yards. Ans. 3243 yards.

2. Find the solidity of a cone whose diameter is 7 ft.

II in., and slant side 16 ft. 1 in.

I

Ans. 255.879 feet.

3. Find the solidity of a cone whose diameter is 13 ft. 6 in., and slant side 18 ft. 4 in. Ans. 813.7.2 feet.

4.

Find the solidity of the frustum of a cone whose top and bottom diameters are 4 and 7 feet, and the depth 8 feet. Ans. 220. II feet. 5. Find the solidity of the frustum of a cone whose top and bottom diameters are 18 ft. 5 in., and 13 ft. 1 in., and slant side 21 ft. 5 in. Ans. 4181.4 feet. 6. The top and bottom diameters of a tub are 4 ft. 3 in. and 2 ft. 9 in., and the depth 4 ft. 10 in.; how many gallons will it hold? Ans. 294.

7. Find the weight of a conical plug of cast iron whose diameter is 9 inches, and height 15 inches.

Ans. 79 lbs.

8. The sides of a right-angled triangle are 1 ft., 2 ft. II in., 3 ft. 1 in.; if it revolve about the hypothenuse, find the solidity and surface of the solid.

Ans. 283 feet, 11143 feet.

THE SPHERE.

O

This solid is formed by the revolution of a circle about

a diameter.

XXXIV.

To find the surface of a sphere.

Multiply the square of the diameter by 34.

Ex. Find the surface of a sphere whose diameter is

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To find the curve surface of the segment of a sphere.

(1) When the diameter of the sphere and the height of the segment are given: Multiply the diameter of the sphere by the height of the segment; then multiply by 34.

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