Volumes of Spheres.-Every sphere may be ht of as composed of a great number of very small pyramids. The volume of each

pyramid is Bh or BR, since eight of the pyramid is the radius e sphere. The volume of the re is the sum of the volumes of hese little pyramids.. Their total mes will be the radius e sphere times the sum he areas of their bases.

? The sum of the areas heir bases is the total face of the sphere. y? Hence,

V=RX 4TR2

(1)

(2) How?

EXERCISES

Explain formula (2) fully.

2. Find the volume of a sphere whose radius is 3"; 6 .; .8 dm.; 1.2 ft.

3. Find the volume of the sphere which has a diameter of n.; 10 cm.; 6 dm.; 1.4 ft.

4. From (2) express R in terms of V

d.

5. Find the radius of the sphere whose olume is 722 cm3.; 1437 cu. in.

6. The lower part of this tank is a emisphere. The cylindrical part is 20 ft. gh. Both have a radius 9 ft. Find its apacity in cubic feet and in gallons.

19'

T

20

IX

DIRECT AND INDIRECT MEASUREMENTS

195. Direct Measurements.-Two kinds of measu ments are made; direct and indirect. Direct measureme are such as (1) finding the length of a house by means of yard-stick or a foot-ruler; (2) finding the weight of a sa of flour or the weight of a load of coal by the use of scal (3) finding the volume of the milk in a pail or the volume the berries in a basket by the use of a pint or a quart mea ure. All measures in which the result is in terms of t unit of measure needed are direct measurements. In mo cases no computations are needed in direct measurements

196. Indirect Measurements.-The plasterer measur the height and the length of the walls he is to plaster. Fro this he computes their areas. He measures a length whe he wishes to find an area. In finding the number of bushe of wheat a load contains, the grain dealer weighs the loa and after the grain has been unloaded, he weighs the wagon The difference between these two weights he divides by 6 the number of pounds a bushel of wheat weighs. Why The grain dealer weighs grain when he wishes to find it volume. These are only two of many indirect measure ments. In fact, most of our measurements are indirect All measurements in which the results are not found directl but are computed from other measurements are indirect Areas and volumes have all been found from linear measure

EXERCISES

State which of the following measurements are direct which are indirect:

Measuring the length of a room;

Weighing a can of milk to find the number of quarts of milk it contains;

Measuring milk with a quart measure;

Weighing out sugar on a pair of scales;

Measuring the length and width of a room to find the size of rug needed for it;

Weighing a load of coal and the empty wagon to find the number of tons of coal;

. Taking the dimensions of a coal-bin to find the number of tons it will hold;

Measuring potatoes with a peck measure;

Measuring a hay-stack to find the number of tons of hay it contains.

2. Suggest three direct measurements other than those ntioned above. Why are they direct?

3. Suggest three indirect measurements other than those entioned above. Why are they indirect?

4. Explain what measurements must be taken and how ese are to be used in finding the volume and the surface of sphere. Are these direct or indirect measurements?

5. A sphere is placed between two blocks as shown in Le picture on page 181. The sides of the blocks nearest the here touch the meter stick at 4.6 cm. and 9.3 cm. from e same end of the meter stick. What is the radius of the here? Find its surface. Find its volume.

How fan de vou

in 10 atong 2 in 1 aton ? Hao this

7. Select two points on the school ground or on th street and estimate the distance between them. Step of this distance to check your estimate. What is the per cen of error of your estimate? Make an estimate of three o four other distances in the same manner and check the es timate by stepping off the distance. Also find your per cen of error in each case.

8. The former army pace was 2 ft. How many inches is this? How many steps did a soldier take in marching a mile of 5280 ft.? 5 mi.? 15 Km.?

9. The new army step is 28 in. How many feet and fraction of a foot is this? How many steps will a soldier take in marching a mile? 5 mi.? 15 Km. ?

10. If a soldier takes 126 of these steps per minute, how far does he go in a minute? in hr. ? in 1 hr. ? in 3 hr. ?

11. An acre is 160 sq. rd. What length and width could a rectangle have so that its area would be an acre? Give at least three different suggestions.

12. Use the most convenient one of the different rectangles suggested for Ex. 11 and lay off a piece of ground which will be an acre.

13. How long are the sides of a square containing one acre?

14. Step off the length and the width of some city block and from this find its area in acres.

15. How many tons are there in a load of coal weighing 5230 lb. and the empty wagon weighing 1350 lb.? Is this a direct or an indirect measurement? Notice that you get weight by weighing.

16. How many bushels of oats are there in a load weighing 2940 lb. if the empty wagon weighs 1190 lb.? What is

the cost of the oats at 57 d nor hushol ?

the body and divide that by the weight of a cubic or a cubic foot of the substance. A cubic inch of cork hs about .15 oz. A piece of cork weighing 2 lb., that O oz., would have a volume of (40.15) cu. in. How y cubic inches is this?

the object is such that it will sink and not dissolve in er, the following is a good plan: Immerse the object in a nder of water and note how high the water rises. The me of the object and the volume of the cylinder into ch the water has risen compare how? For instance, a x is immersed in a cylinder whose radius is 2" in which water stands 8" above the bottom. After the rock is in water the top level is 11" above the bottom. How high the rock raise the water in the cylinder? What is the ume of the rock?

EXERCISES

1. A cubic foot of ice weighs 57.5 lb. What is the volhe in cubic feet of a piece of ice weighing 230 lb. ?

2. How many cubic feet of ice are there in a piece eighing 500 lb.? Give the result correct to two decimal

aces.