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Question 933877: the future value of $5000 invested for 4 years at rate r is compounded annually, is given by S=5000(1+r)^4. a) graph with these points [0,0.24] by [0,12000]-i did this step on the calculator. b) use the root method to find the rate r, as a percent for which the future value is $10,368. r=.2 c)What rate as a percentage gives 2320.50 in interest on this investment?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the future value of $5000 invested for 4 years at rate r is compounded annually, is given by S=5000(1+r)^4.
a) graph with these points [0,0.24] by [0,12000] - i did this step on the calculator.
i'm not sure what they are asking for in this question.
looks like you already answered it.
b) use the root method to find the rate r, as a percent for which the future value is $10,368. r=.2
present value is 5000
future value is 10,368
rate = .2
f = p * (1 + r) ^ n
f = future value
p = present value
r = interest rate per time period
n = number of time periods
time period is years.
(1 + r) = (1 + .2) = 1.2
equation becomes:
10368 = 5000 * (1.2)^n
divide both sides of this eqaution by 5000 to get:
10368 / 5000 = 1.2^n
simplify to get:
2.0736 = 1.2^n
take the log of both sides of this eqution to get:
log(2.0736) = log(1.2^n)
since log(1.2^n) = n*log(1.2), this equation becomes:
log(2.0736) = n * log(1.2)
divide both sides of this eqution by log(1.2) to get:
log(2.0736) / log(1.2) = n
solve for n to get:
n = 4
that's your solution.
plug it into your original equation to get:
10368 = 5000 * (1.2)^4
solve the equation to get:
10368 = 10368
the solution is confirmed as good.
c)What rate as a percentage gives 2320.50 in interest on this investment?
your interest desired is 2320.50
this means your future value desired is 5000 + 2320.5 which is equal to 7320.5
you are looking for an interest rate that gets you a future value of 7320.5
your equation is f = p * (1 + r)^n
f = 7320.5
p = 5000
n = unknown
r = unknown
you have 2 unknowns here which doesn't allow you to come up with a single solution.
if you know one of the unknowns, you can solve for the other unknown.
the equation becomes:
7320.5 = 5000 * (1 + r)^n
divide both sides of this equation by 5000 to get:
7320.5 / 5000 = (1 + r)^n
if you know n, then you use the ratio method described above.
if you know r, then you use the log method described above.
for example:
if n = 1, then the equation becomes:
7320.5 = 5000 * (1 + r)^1
divide both sides of the equation by 5000 and you get:
7320.5 / 5000 = (1+r)^1 which becomes:
7320.5 / 5000 = 1 + r
subtract 1 from both sides of this equation to get:
7320.5 / 5000 - 1 = r
solve for r to get:
r = .4641 which is equal to 46.41%.
5000 * 4.4641 = 7320.5
the solution is good.
for another example:
if r = .09, then your equation becomes:
7320.5 = 5000 * (1.09)^n
divide both sides of this equation by 5000 to get:
7320.5 / 5000 = (1.09)^n
take the log of both sides of this equation to get:
log(7320.5/5000) = log(1.09^n)
since log(1.09^n) = n * log(1.09), the equation becomes:
log(7320.5/5000) = n * log(1.09)
divide both sides of this equation by log(1.09) to get:
log(7320.5/5000) / log(1.09) = n
solve for n to get:
n = 4.423890819
replace n with 4.423890819 in your original equation to get:
7320.5 = 5000 * (1.09)^4.423890819
solve the equation to get:
7320.5 = 7320.5
the solution is good.
if you know r, you can solve for n.
if you know n, you can solve for r.
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