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before extracting the square roots, the adfected quadratic would have given the same two roots.
E X A M P L E III.
Let a stone be dropt into an empty pit; and
let the time from the dropping of it to the hearing the found from the bottom be given: To find the depth of the pit.
Let the given time be a; let the fall of a heavy body in the ist second of time (16.122 feet) be b; also, let the motion of found in a second (1142 feet) be c.
Let the time of the stone's fall be 14 The time in which the found of it moves to the top is
2'a-X The descent of a falling body is
as the square of the time, there-
(1*:** :: b :)
4.Ca-cx Therefore 3 and 4
This equation being resolved, gives the value of x, and from it may
be camcx, the depth of the pit,
got bx? or
If the time is 10", then x=8.885 nearly, and the depth is 1273
feet. There are several circumstances in this problem which render the conclusion inaca curate.
1. The values of c and b, on which the folution is founded, are derived from experiments, which are subject to confiderable inaccuracies.
2. The resistance of the air has a great effea in retarding the descent of heavy bodies, when the velocity becomes so great as is supposed in this question ; and this circumstance is not regarded in the folution.
3. A small error, in making the experiment to which this question relates, prodúces a great error in the conclusion. This circumstance is particularly to be attended to in all physical problems; and, in the present case, without noticing the preceding imperfections, an error of half a second, in estimating the time, makes an error of above 100 feet in the expression of the depth of the pit.
iii. Of Interest and Annuities.
The application of algebra to the calculation of interests and annuities, will furnish proper examples of its use in business. Algebra cannot determine the propriety or justice of the common fupposition on which these calculations are founded, but only the necessary conclusions resulting from them.
In the following theorems, let denote any principal sum of which 1 l. is the unit, t the time during which it bears interest, of which 1 year shall be the unit, » the rate of interest of 1 l. for 1 year, and let s be the amount of the principal sum p with its interest, for the time t, at the rate r.
I. Of Simple Interest.
s=p+ptr, and of these four, s, p, t, r, any three being given, the fourth
be found by resolving a simple equation.
The foundation of the canon is
oba vious; for the interest of il. in one year is
years it is tr, and for p pounds it is pir; the whole amount of principal and interest muft therefore be ptptr=s.
II. Of Compound Interesi.
When the simple interest at the end of every year is supposed to be joined to the principal fum, and both to bear interest for the following year, money is said to bear compound interest. The same notation being used, let itr=R. Then s=pR'.
For the simple interest of 1 l. in a year is r, and the new principal sum therefore which bears interest during the second year is i tr=R; the interest of R for a year iš rR, and the amount of principal and interest at the end of the 2d year is R+rR= Rxitr=R2. In like manner, at the end of the 3d year it is R, and at the end of t years
it is R', and for the sum p it is PR'=s. Cor. 1. Of these four p, R, t, s, any three being given, the fourth may be found.
When t is not very small, the solution will be obtained most conveniently by logarithms. When R is known r may be found, and conversely.
Ex. If 500l. has been at interest for 2 i years, the whole arrear due, reckoning 4 per cento compound interest, is 1260.12 l. or 1260 1. 2 s. 5 d. In this case, p= 500, R=1.045, and t=21, and s=1260.12, and any one of these may be derived by the theorem from the others being known. Thus, to find s; I.R'=tXI.R=21 X 0.0191163 =0.4014423, therefore R'=2.52042, and s= (PR=) 500 X 2.520242=1260.121.
Cor. 2. The present worth of a fum (s) in reversion that is payable after a certain time t is found thus. Let the present worth be x, then this money improved by compound interest during i produces xR', which ipust be equal to s, and if xR'=s, x=
R The time in which a sum is doubled at compound interest will be found
1.2. thus, PR'=2p, and R=2, and t=
1.R. thus, if the rate is 5 per cent, r=.05, and