Questions - Continuous Compounding
Questions/Answers....Continuous Compounding.
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> 6.EQUATIONS":
>
> Can someone help me? Determine the sum accumulated in 20years by each of
> the following investment programs. Suppose that the interest is 9.5%
> compounded continuously and that the deposits are also made continuously. a)
> Nothing initially and $1500 per year for 20 years b) $10,000 initially and
> $1000 per year for 20 years c) $20,000 initially and $500 per year for
> 20years d) $30,000 initially and no additional deposits Note that in each
> case the total amount deposited is $30,000 Thanks
> ------------------------------------------------------------------------
ANSWER
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formula for continuous compounding is
amount = p*e^(nr).....where p=principal.....n= number of
years....r=interest rate in fraction.
in case of periodic investment the formula for continuous compounding is
IMPORTANT:-YOUR WRITE UP GIVES THE IMPRESSION THAT THE INITIAL
INVESTMENT IS AT THE BEGINING OF THE 20 YEAR PERIOD,WHERE AS THE
PERIODIC INVESTMENTS START AT THE BEGINING OF NEXT YEAR.THIS IS VERY
IMPORTANT AND MAKES A DIFFERENCE.NORMALLY,THE PERIODIC INVESTMENTS
ALSO START AT THE BEGINING OF EACH YEAR.PLEASE CHECK AND INFORM IF THE
ASSUMPTION IS CORRECT OR NOT.THE FOLLOWING SOLUTION IS BASED ON THAT
ASSUMPTION.
amount = y*[e^(n-1)r + e^(n-2)r + e^(n-3)r +...............+ e^2r +
e^1r + e^0r]...
this is g.p.with first term = 1 , common ratio = e^r and number of
terms = 20.its sum is
amount = y*[e^(nr) - 1 ] /[e^(r) -1]
IF INSTALMENTS ALSO START FROM DAY ONE OF THE 20 YEAR PERIOD THEN THE FORMULA IS
AMOUNT = Y*[ E^NR+ E^(N-1)R + E^(N-2)R +...........+ E^2R + E^1R ]
THIS IS A G.P. WITH FIRST TERM = E COMMON RATIO = E^R
AND NUMBER OF TERMS =20
AMOUNT = Y*E[E^(NR) - 1 ] / [E^(R) - 1 ]
hence for the four cases given the answer is as follows
X=initial= 0 10000 20000 30000
periodicY=1500 1000 500 0
amount=(X*(e^(nr)))+Y*((e^(nr)
amount= 85580.4 123912.5 162244.7 200576.8
col.1 is number of years for the investment to earn interest.col.2,3,4,5 are the 4 alternatives.
20 0 66858.9 133717.9 200576.8
19 9119.957173 6079.971449 3039.985724 0
18 8293.442216 5528.961478 2764.480739 0
17 7541.831885 5027.887923 2513.943962 0
16 6858.337793 4572.225195 2286.112598 0
15 6236.786764 4157.857843 2078.928921 0
14 5671.565081 3781.043388 1890.521694 0
13 5157.567781 3438.378521 1719.18926 0
12 4690.152548 3126.768365 1563.384183 0
11 4265.097785 2843.398524 1421.699262 0
10 3878.564489 2585.709659 1292.85483 0
9 3527.061571 2351.374381 1175.68719 0
8 3207.414331 2138.27622 1069.13811 0
7 2916.735782 1944.490521 972.2452607 0
6 2652.400577 1768.267051 884.1335257 0
5 2412.021296 1608.014197 804.0070987 0
4 2193.426884 1462.284589 731.1422947 0
3 1994.643042 1329.762028 664.8810141 0
2 1813.874396 1209.249598 604.6247988 0
1 1649.488283 1099.658855 549.8294276 0
0 1500 1000 500 0
TOTAL AMOUNT AT THE END OF 20 YRS.
85580.36968 123912.5242 162244.6787 200576.8333
posted by mathsmanagementcommonsense at 9:35 AM
4 Comments:
Can someone help me with the question below? Seems long, but they are straightforward answers.
Question) Classify the following equations - linear, nonlinear, seperable, exact, homogeneous, or one that requires an integrating factors as follows: . If an integrating factor is required, specifiy it, BUT DO NOT SUBMIT SOLUTIONS TO ANY OF THE EQUATIONS BELOW.
(a) dy/dx = (x^3 -2y) / x
(b) (x + y)dx - (x -y)dy = 0
(c) dy/dx = (2x + y)/ (3 + 3y^2 - x)
(d) (x + e^y)dy - dx = 0
(e) dy/dx = - (2xy + y^2 + 1)/ (x^2 + 2xy)
(f) x (dy/dx) + xy = 1 -y
(g) dy/dx = x/ (x^2 y + y^3)
(h) x (dy/dx) + 2y = sin(x)/x
(i) dy/dx = - (2xy + 1)/ (x^2 +2y)
(j) (3y^2 + 2xy)dx - (2xy + x^2)dy = 0
(k)(x^2 + y)dx + (x + e^y)dy = 0
(l) (dy/dx) + y = 1/(1+e^x)
(m) xdy - ydx = (xy)^(1/2) dx
(n) (x+y)dx + (x+2y)dy = 0
(o) (e^x + 1) dy/dx = y - ye^x
(p) dy/dx = (x^2 + y^2)/ (x^2)
(q) dy/dx = e^2x + 3y
(r) (2y + 3x)dx = -xdy
(s) xdy - ydx = 2x^2y^2dy
(t) y' = e^(x+y)
10:35 AM
Question) Classify the following equations - linear, nonlinear, seperable, exact, homogeneous, or one that requires an integrating factors as follows: . If an integrating factor is required, specifiy it, BUT DO NOT SUBMIT SOLUTIONS TO ANY OF THE EQUATIONS BELOW.
(a) dy/dx = (x^3 -2y) / x
(b) (x + y)dx - (x -y)dy = 0
(c) dy/dx = (2x + y)/ (3 + 3y^2 - x)
(d) (x + e^y)dy - dx = 0
(e) dy/dx = - (2xy + y^2 + 1)/ (x^2 + 2xy)
(f) x (dy/dx) + xy = 1 -y
(g) dy/dx = x/ (x^2 y + y^3)
(h) x (dy/dx) + 2y = sin(x)/x
(i) dy/dx = - (2xy + 1)/ (x^2 +2y)
(j) (3y^2 + 2xy)dx - (2xy + x^2)dy = 0
(k)(x^2 + y)dx + (x + e^y)dy = 0
(l) (dy/dx) + y = 1/(1+e^x)
(m) xdy - ydx = (xy)^(1/2) dx
(n) (x+y)dx + (x+2y)dy = 0
(o) (e^x + 1) dy/dx = y - ye^x
(p) dy/dx = (x^2 + y^2)/ (x^2)
(q) dy/dx = e^2x + 3y
(r) (2y + 3x)dx = -xdy
(s) xdy - ydx = 2x^2y^2dy
(t) y' = e^(x+y)
2:41 PM
Help please!
Solve the following equation, then write solution in set notation.
|x+10|<5
8:57 AM
OOps forgot one more.
For the polynomial f(x)=-2x^3+3, which statement is true?
a)as x goes to minus infinity, f(x) is increasing; and as x goes to infinity, f(x) is decreasing.
b)as x goes to minus infinity, f(x) is decreasing; and as x goes to infinity, f(x) is decreasing
c) neither
Thank you!
9:01 AM
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