Exercise 51

1. Find the volume of a cone, the radius of the base being 4 in. and the altitude 10 in.

2. Find the volume of an ice cream cone if the diameter of the base is 24 in. and the altitude 4 in.

3. How many square yards of canvas are required to make a conical tent 10 ft. in diameter and 9 ft. high, allowing 1 sq. yd. extra for seams and waste?

4. A conical roof has a diameter of 20 ft. and a slant height of 16 ft. Find its area.

5. Find the total surface of a cone if the radius of the base is 5 in. and the slant height is 22 in.

6. What is the lateral surface of an ice cream cone, the radius of the base being 1 in. and the slant height 4 in.?

7. A cone and two pyramids have the same slant height, 24 in. The base of the cone is 8 in. in diameter; the base of one pyramid is 8 in. square; the base of the other pyramid is a regular hexagon 4 in. on a side. Find the lateral surface of the cone and of each of the pyramids. Which has the largest lateral surface? Which the smallest?

8. Find the altitudes of cones if the slant heights and the radii of the bases are the following numbers respectively: 5,4; 10, 6; 13, 5; 20, 16; 17, 8; 2, 1.

9. How many cubic yards of sand in a pile, assumed to be a cone, 12 ft. in diameter and 4 ft. high?

10. A brass paper weight is in the form of a cone. The radius of the base is 1 in. and the altitude is 2 in. How many such paper weights can be made from a cubic foot of brass?

49. Volume and surface of a sphere. A sphere is a solid bounded by a curved surface every point of which is equally distant from a point within called the

center.

Let a sphere be cut into hemispheres. Cover the curved surface and also the flat surface of a hemisphere with a cord, Figure 66. By comparing the length of the cord required to cover the curved surface with that required to cover the flat surface it will be seen that the curved surface of a hemisphere is twice the flat surface of the hemisphere. Since the area of the flat surface of the hemisphere is πr2,

we have

S=4 πr2,

where S is the whole surface of the sphere.

Sphere

FIG. 65

Rule. The area of a sphere equals four times π times the square of the radius.

It is proved in geometry that the formula for the volume of a sphere is Vr3, where V is the volume, and r is the radius of the sphere.

Exercise 52

1. How many square inches in the surface of a globe that is 6 in. in diameter?

2. How many pounds of steel are required to make 1000 steel ballsin. in diameter? The specific gravity of steel is 7.83.

3. Find the volume of the earth, assuming it to be a sphere 8000 mi. in diameter.

4. Find the volume of the largest sphere that can be cut from a 10-inch cube.

5. Find the volume of the largest cylinder that can be cut from a 10-inch cube.

Exercise 53. Problems about concrete work

In the following problems 1 sack of cement is to be counted as 1 cu. ft. A 1:2:3 mixture of cement, sand, and gravel means a mixture of 1 part of cement, 2 parts of sand, and 3 parts of gravel. Use the following prices: Cement, $.40 a sack; sand and gravel, $1.10 a cubic yard; cinders, $.25 a cubic yard. It will be assumed that if a fraction of a sack of cement or a fraction of a cubic yard of the other materials is needed, a whole sack, or a whole cubic yard, will be bought.

1. Find the cost of the materials to make a concrete watering trough 8 ft. long, 4 ft. wide, and 3 ft. deep, inside measurements, with sides and bottom 6 in. thick, made of a 1:23 mixture of cement, sand, and gravel.

2. Find the cost of the materials for one mile of concrete sidewalk, 4 ft. wide, which has a base of cinders 4 in. thick, then a layer of a 1:7 mixture of cement and gravel 3 in. thick, and a top in. thick of a 1 : 2 mixture of cement and sand.

3. Find the cost of the materials for 1000 fence posts, 8 in. by 4 in. at the bottom, 4 in. square at the top, and 7 ft. long. They are made of a 1:3 mixture of cement and gravel, and each post is reënforced by 4 wires running through it lengthwise, each wire being 4 in. longer than the post. This wire costs 2.5¢ a pound. Twenty feet weigh a pound.

4. Find the cost of the materials for making 10 concrete gate posts 12 in. square at the bottom, 12 in. by 8 in. at the top, and 8 ft. long. They are made of a 1:3 mixture of

cement and gravel and each is reënforced by 4 steel rods costing 4¢ a foot.

5. What is the cost of the materials for the posts needed to fence a lot 132 ft. by 206 ft., placing the posts a rod apart, and having a gate 8 ft. wide in each of the long sides of the lot? The posts used are of the sizes given in problems 3 and 4.

6. An excavation 57 ft. 9 in. wide and 86 ft. 3 in. long is made for a building. The excavation is 5 ft. deep except for a trench 2 ft. wide running around the outside of the excavation, which is 8 in. deeper. Find the cost of making the excavation at 40 cents a cubic yard.

7. The trench, in the previous problem, is for the base, or footing, of the foundation of the building. This footing is 2 ft. wide and 8 in. deep, and is made of a 1 : 6 mixture of cement and gravel. Find the cost of the materials for the footing.

8. The foundation for this building is 84 ft. 7 in. long and 56 ft. 1 in. wide, outside measurements, and is 10 in. thick and 8 ft. 9 in. high. It is made of a 1:3: 5 mixture of cement, sand, and gravel. Find the cost of the materials.

9. The basement of this building has a concrete floor which cost 10 cents a square foot. Find its cost.

10. A circular grass plot 120 ft. in diameter is surrounded by a concrete walk 6 ft. wide. This walk is constructed in the same way as the one in Problem 2. Find the cost of the materials.

11. What is the cost per square foot of the materials for the walk in the previous problem? Find the answer correct, to .01 cent.

12. If the circular grass plot of problem 10 had twice as great a diameter, what would be the cost of the materials for the walk? How many times as much as for the plot whose diameter is 120 ft.?

Exercise 54. Problems about silos Silos are used to store food for livestock. of a silo are called silage.

The contents

Corn is much used for silage. The corn is cut green,

FIG. 67

run

through a cutter, and run into the silo. The silage is tramped down as tightly as possible so as to shut out the air. Silos are usually cylindrical in shape.

If silage is exposed too long to the air, it becomes unfit for feed. Hence in building a silo care is taken to make it the proper size so that silage will be removed fast enough to prevent decay. The surface of the silage that is exposed to the air, which is the same as the area of one end of the silo, is called the "feeding surface."

1. A cylindrical silo is 18 ft. in diameter, inside measure, and 30 ft. high. Find its capacity in cubic feet.

2. If a cubic foot of corn silage weighs 40 lb., how many tons of corn silage will the silo in the previous problem contain?

3. How many cubic feet of corn silage in 100 tons?

4. How long will 100 tons of silage last 40 cows if each cow eats 25 pounds a day?

5. It is estimated that there should be about 6 square feet of feeding surface for each cow fed. How many square feet of feeding surface are needed for 25 cows? What is the diameter of a cylindrical silo that has this amount of feeding surface, correct to the nearest foot?

6. Silos are usually made with inside diameters of 12, 14, 16, 18, 20, or 22 ft. Which is the smallest of these sizes that