« ForrigeFortsæt »
before extracting the fquare roots, the adfected quadratic would have given the fame two roots.
Let a ftone be dropt into an empty pit; and let the time from the dropping of it to the bearing 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 1ft fecond of time (16.122 feet) be b; alfo, let the motion of found in a fecond (1142 feet) be c.
Let the time of the ftone's fall be 1x
moves to the top is
The defcent of a falling body is
The depth from the motion of
found is alfo
Therefore 3 and 4
This equation being
value of x, and from it
refolved, gives the
may be got bx or
ca-cx, the depth of the pit.
If the time is ro", then x 8.885 near
ly, and the depth is 1273
There are feveral circumftances in this problem which render the conclufion inac
1. The values of c and b, on which the folution is founded, are derived from experiments, which are fubject to confiderable inaccuracies.
2. The resistance of the air has a great effect in retarding the descent of heavy bodies, when the velocity becomes so great as is fuppofed in this question; and this circumstance is not regarded in the folution.
3. A fmall error, in making the experiment to which this question relates, produces a great error in the conclufion. 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 fecond, in estimating the time, makes an error of above 100 feet in the expreffion of the depth of the pit.
III. Of Interest and Annuities.
The application of algebra to the calculation of interefts and annuities, will fur nish proper examples of its use in business. Algebra cannot determine the propriety or juftice of the common fuppofition on which thefe calculations are founded, but only the neceffary conclufions refulting from them.
In the following theorems, let p denote any principal sum of which 1 1. is the unit, t the time during which it bears interest, of which 1 year fhall be the unit, the rate of intereft of 1 1. for 1 year, and let s be the amount of the principal fump with its intereft, 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 may be found by refolving a fimple equation.
The foundation of the canon is very obvious; for the intereft of 11. in one year is * for.t years it is tr, and for p pounds it is pir; the whole amount of principal and intereft must therefore be p+ptr=s.
II. Of Compound Intereft.
When the fimple intereft at the end of every year is fuppofed to be joined to the principal fum, and both to bear intereft for the following year, money is faid to bear compound intereft. The fame notation being used, let 1+r=R. Then s=pR'..
For the fimple intereft of 1 1. in a year is r, and the new principal fum therefore which bears intereft during the fecond year is trR; the intereft of R for a year is rR, and the amount of principal and intereft at the end of the 2d year is R+rR Rx+r=R2. In like manner, at the end of the 3d year it is R3, and at the end of t years it is R', and for the fum p it is pR' Cor. 1. Of thefe four p, R, t, s, any three being given, the fourth may be found.
When is not very small, the folution will be obtained moft conveniently by logarithms. When R is known ʼn may be found, and converfely.
Ex. If 500l. has been at interest for 21 years, the whole arrear due, reckoning 4 per cent. compound intereft, is 1260.12 l. or 1260k 2 s. 5 d. In this cafe, 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. to find s; l.R'=txl.R=21X0.0191163 0.4014423, therefore R'=2.52042, and = (pR'=) 500X2.520242=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 prefent worth be x, then this money improved by compound intereft during produces R', which must be equal to s, and if xR's, x=
Cor. 3. The time in which a fum is doubled at compound intereft will be found thus, pR' 2p, and R'=2, and t=
thus, if the rate is 5 per cent, r=.05, and