1) Determine how much you
will have to save each month at 3, 6, 9, and 12 percent compounded
monthly...

Question

1) Determine how much you
will have to save each month at 3, 6, 9, and 12 percent compounded
monthly for you to accumulate a nest egg for retirement. The
variables are current age, age of retirement, nest egg size, and
interest rate. Show all work and cite as appropriate.

2) Measure the height of a
tree or a street light utilizing trig. Report what you did and how
you did it, including all measurements.

3) The kinetic energy of a
vehicle is described by KE(v) = (.5*mass)v^2, where V is velocity.
Graph the function. What is the percent energy increase if we go from
30 to 40 mph? From 40 to 50? 50 to 60? 60 to 80? How can we provide a
safer driving environment and how can we reduce fuel costs? Is there
a conflict between the two in all regards, or just one?

David · Accepted Answer

Question 1: Determine how much you will have to save each month at 3, 6, 9, and
12 percent compounded monthly for you to accumulate a nest egg for retirement. The
variables are current age, age of retirement, nest egg size, and interest rate. Show
all work and cite as appropriate.
Variable definitions:
ca = current age
ra = retirement age
nest egg size = N
interest rate = r
Total Amount after compounding interest = A
A = P
(
1 + r
n
)nt
where
A = Final Amount = N (nest egg size)
P = Principal Amount or Initial Investment
r = Annual Interest Rate
n = no. of times interest is compounded per year = 12
t = no. of yrs amount is deposited = ra-ca
Let k = nt
Calculations:
Final Amount from 1st month’s deposit = P ×
(
1 + r
12
)k
Final Amount from 2nd month’s deposit = P ×
(
1 + r
12
)k−1
...
Total Amount at the end = P
(
1 + r
12
)k
+ P
(
1 + r
12
)k−1
+ . . . P
(
1 + r
12
)1
= [P
(
1 + r
12
)k
+ P
(
1 + r
12
)k−1
+ . . . P
(
1 + r
12
)1
+ P ] − P
Let
(
1 + r
12
)
= λ
Total Amount at the end = [Pλk + Pλk−1 + · · · + Pλ+ P ] − P
= P
(
λk+1−1
λ−1
)
− P
= Pλ
(
λk−1
λ−1
)
= N
Hence,
P =
(
N(λ−1)
λ(λk−1)
)
For r = 3 %,
λ = 1.0025
P =
(
N(2.49×10−3)
1.0025k−1
)
For r = 6 %,
1
λ = 1.005
P =
(
N(4.975×10−3)
λk−1
)
For r = 9 %,
λ = 1.0075
P =
(
N(7.4×10−3)
λk−1
)
For r = 12 %,
λ = 1.01
P =
(
N(9.9×10−3)
λk−1
)
References:
http://qrc.depaul.edu/StudyGuide2009/Notes/Savings%20Accounts/Compound%
20Interest.htm
http://en.wikipedia.

1) Determine how much you will have to save each month at 3, 6, 9, and 12 percent compounded monthly for you to accumulate a nest egg for retirement. The variables are current age, age of retirement,...

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