14. The members of a society each contribute the same amount to a fund of $336. If there were three members less, each member would have to contribute $2 more. Find the number of members.

15. A person purchased a certain number of sheep for $175; after losing two of them he sold the rest at $2 a head more than he gave for them, and by so doing gained $5 by the transaction. Find the number of sheep purchased.

16. A cask contains 360 gallons of wine; a certain quantity is drawn and an equal quantity of water is put in; from this mixture the same quantity as before is drawn, and 84 gallons in addition; on replacing the drawn liquid with water it is found that the barrel contains equal quantities of wine and water. How many gallons were drawn the first time?

17. A number of men pass a certain time in a hotel and on leaving they have a bill of $12 to pay. Had there been 4 more in the party and had each spent 25 cents less, their bill would have been $15. What was the number of men?

18. A merchant paid a certain sum for a horse, later he sold the horse for $144 and thereby gained as much per cent as the horse cost him dollars originally. What did he pay for the horse?

19. What is the quotient, whose dividend is n times smaller than its divisor, and the sum of the quotient and its reciprocal is n?

20. A manufacturer had agreed to pay a capitalist $8,800 after 7 months and $5,940 at the end of 1 year. After how many months can the manufacturer pay back the capitalist the total amount, $14,740, if interest at 5% per annum is charged for the money which he paid later than it was due and if a rebate of 5% per annum is allowed for the money paid before it was due?

21. A capitalist lent k dollars at a certain rate per cent and withdrew each year 6 dollars; at the end of 2 years there remained k' dollars invested. At what per cent was the money lent?

PROBLEMS CONNECTED WITH THE THEOREM OF PYTHAGORAS

419. In this section only those problems will be discussed which are connected with the Theorem of Pythagoras for rightangled triangles; first because it is not desired to make the discussion too extended, and secondly because the Theorem of Pythagoras for acute- and obtuse-angled triangles has more of a trigonometric interest.

The first of the topics just mentioned is the most important, because all problems connected with the second topic depend for their solutions upon the first and because the larger part of physical and technical problems are connected with the relation discussed in the first topic.

We recall that in any right-angled triangle ABC, Fig. 2, by Geometry:

PROBLEM I.—In a right-angled triangle the difference between the longer and shorter legs is equal to the difference between the hypotenuse and the longer leg. How long are the sides of the triangle, if the given difference is 2 inches?

Then it follows from 2419, formula (1), that one has the equation

Since the quantities introduced into the equation have the designation "inches", we obtain as the length of the sides of the triangle:

The solution

=

- 2, has no interpretation, inasmuch as it is assumed in the hypothesis of the problem that any one side is measured in but one direction, namely positive direction.

PROBLEM II.-A chord is drawn through a point P, which is 13 inches from the center of a circle of radius 15 inches. The chord is divided by P into two segments, one of which is 10 inches longer than the other. How long is the chord?

(2) OC=1 169 25 12.

Accordingly the equation of condition will be, from (1) and (2), (3) V 15(c+5)2 = 12. Hence,

(x+5)= 81,

chord AB been introduced as the unknown quantity, the determining equation would have been a pure quadratic.

Here, as in Problem I, the negative solution

- 14 has no meaning excepting that it is a solution of the same equation of which 4 is a root.

=

PROBLEM III.—An isosceles triangle whose sides are in the ratio a: a b is so inscribed in a square whose side is p inches long that the vertex of the triangle coincides with one corner of the square and the vertices at the base of the triangle lie on the sides of the square opposite to the common vertex. How long are the sides of the triangle?

Solution. Since the sides of the triangle sought, AEF, Fig. 4, are in the ratio aab, AE can be represented by ax, AF by ax, and EF by bx. Since AB P,

for BE substitute the expression

::.

b2x2 = ( p −1 a2x2 − p2)2 + (p −√ a2x2 — p2)2,

b2x2 = 2 p2 + 2 a2x2 - 2 p2 - 4 pV a2x2 — p2,

2 a2).c2 = 4 p↓ a2x2 — p2,

(b2 — 2 a2)*x*= 16 p2(a2x2 — p2),

(b2 2 a2)*x*- 16 a2p2x2 + 16 p1 = 0.

(b2 — 2 a2)2x2 = 8 «2p2 ± 1 64 a'p' — 16 p1(b2 — 2 a2)2

2 a2± b 1 4 a2. b2.

x with the positive* sign, the side of the

PROBLEM IV.-If the corners of a square are cut off so that a regular octagon remains, how long is a side of the latter?

Let the portion cut off from one end of a side of the square, Fig. 5; then a 2 x is equal to a side AB of the octagon, and

A

a

B

P

Since the octagon falls within the square, the lower sign must be taken, for if the upper signs were taken the side of the octagon would be longer than the side of the square, which is not possible.

*Since any side of a triangle is less than the sum of the other two sides, 2 a > b, and 4 a2 >> b2; hence v 4-b is always real.

PROBLEM V.-The median lines drawn from the vertices at the acute angles of a right-angled triangle to the perpendiculars are a and b. How long are the sides of the triangle?

Hence, it follows from the triangle ACE (Fig. 6) that

6. If a perpendicular of a right-angled triangle is 11 inches in length and is prolonged beyond the hypotenuse 1 times the length of the other perpendicular, and the point thus determined joined to the other extremity of the hypotenuse, a second rightangled triangle is thus constructed, whose hypotenuse is equal to the hypotenuse of the original triangle plus of its unknown perpendicular. How long is the perpendicular in question?

is b?

7. How high is an isosceles triangle whose base is a and side Ans. x± = √ 4 b2 — a2.

8. Two parallel chords are drawn in a circle of radius 25 cm. One of the chords is 14 times as far from the center as the other. If the shorter is 16 mm. shorter than the other, how long is each?