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EXERCISES By means of the principle in § 111 test the correctness of each of the following proportions. 1. g=45
5. 19 : 21 =42: 105. 2. 6:7= 18:21.
6. 18:19=93: 94.
7. –4:-5: : 16:20.
11. (a? — 62) : (a+b)2: :(a−b):(a+b). [Hint. See Formula V, p. 101.]
19 22+10 x+16 _ x+2
4. x2+4 x– 322 -4 By means of the principle in § 111 find the value which x must have in each of the following proportions. 12 X 1
14. 26 - 13 (Hint. We must have
20. 14: x=28: 21. 12.1=x · 3.]
21. (2-5):4: :2:3. 14. 4x 14. 7 = 10
(Hint. We must have
4. 2= (x–5) · 3.] 15. 15: x= 12:8.
22. x:(x-6)=9: 14. 16. 9:15=18:x.
x-3_5 25 8
x-46 1. 32 = x 18. x:31=33: 68. 25. What number bears the same ratio to 2 as 8 does to 3?
[HiNt. Let x represent the unknown number and form a proportion. Solve for x.]
26. What number bears the same ratio to 7 as 2 does to 3? 27. Divide 35 into two parts whose ratio shall be . [Hint. Let u be one part. Then 35-x will be the other part.]
28. Divide 25 into two parts such that the greater increased by 1 is to the lesser decreased by 1 as 4 is to 1.
29. Two men divide $6300 between them so that the parts are to each other in the ratio 3:4. How much does each receive?
30. A man's income from two investments is $850. The two investments bear interests which are in the ratio of 6 to 8. What income does he receive from each ?
31. Concrete for sidewalks is a mixture made of two parts of sand to one part of cement. How much of each is required to make a walk containing 500 cubic feet?
32. Find the number which, when added to each of the numbers 1, 2, 4, and 7, will give four numbers in proportion.
33. Prove that no four consecutive integers, as n, n+1, n+2, n+3, can form a proportion.
34. A bubble of air of volume v units when rising from a depth of d feet below the surface of the water gradually expands until it reaches a volume of V units at the surface such that
V 34 Whence, find the volume at the surface of a spherical bubble which starts at a depth of 100 feet with a radius of 1 inch. (Hint. See Ex. 28, p. 23.]
35. Solve for d in the formula of Ex. 34 (see § 107) and use your result to answer the following question: From what distance below the surface must a bubble rise in order that its volume may increase from 3 cu. in. to 20 cu. in. ?
112. The Lever. If a 2-pound weight be attached to one end of a yardstick and a 1-pound weight to the other end, and the whole be then exactly balanced, as shown in
Figure 46, we have an example of a lever. The point (pivot) around
which the balance Fig. 46.
takes place is called
the fulcrum. If this experiment be tried (and it easily can be at home or in the classroom) it will be found that when the balance is exact, the fulcrum is just 24 inches from one end of the stick, and 12 inches from the other end. Thus the ratio of these two distances is 14, or , which is therefore just the same as the ratio of the two weights to each other.
This experiment illustrates a general law as
Fig. 47 follows :
LAW OF THE LEVER. If two weights W and w are balanced at the ends of any uniform bar at distances D and d respectively from the fulcrum (Fig. 47), we have
EXERCISES — THE LEVER 1. In the figure above suppose W=4 pounds, w=2 pounds, and D=6 inches. What must be the value of d?
[Hint. By the law stated above, we have 4/2 = d/6. Solve for d.]
2. Fill in each of the following question marks (?) in such a way that the balance will be perfect in the above figure.
(a) W=9 pounds, w=3 pounds, d=1 inch, D= ? (6) W=8 ounces, w=4 ounces, d= ?, D=1 foot. (c) W = ?, w=11 pounds, d=1 foot, D=10 inches. (d) W=3 ounces, w=1 pound, d=12 centimeters, D= ?
3. Where must we place the fulcrum under a 12-foot plank in order that a 56-pound boy at one end may balance a 112-pound boy at the other end?
[Hint. Let x be the distance from one end. Then the distance from the other end will be 12-x.]
4. Two boys balance at seesaw on a 12-foot plank. The fulcrum is 5 feet from the heavier boy, who weighs 105 pounds. How much does the other boy weigh?
5. Sometimes, instead of having two weights balanced, we have a single weight balanced by a force, or, as it is usually called, a power. This may happen in several ways as indicated by the following figures.
Note that in the last two figures the fulcrum is at one end of the bar.
In all these cases, if we let W represent the weight, p the power, D the distance of the weight from the fulcrum, and d the distance of the power from the fulcrum, we have
W_d, or WD=pd.
This is called the general law of the lever.
By means of this law, answer the following question. If the fulcrum of a 5-foot crowbar is placed 1 foot from the end, what weight can be lifted by a man weighing 160 pounds ?
(Hint. Here we have Fig. 48A with W= ? p=160 lb., D=1 ft., d=4 ft.]
6. Figure 49 represents a simple form of pump. Suppose that the pump-handle AF is 11 ft. long, while the piston-arm FC is 5 inches long. What will be the upward
pull on the piston at C when there is a downward pressure of 10 pounds at A?
7. Figure 50 represents another common form of pump. With AF and FC measuring the same as in Ex. 6, what will
be the upward pull at C when there is a 10-pound upward pull at A?
[Hint. See Ex. 5, Fig. 48B.]