Growth Periods - Continuous Growth

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mexx

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please help me solving this...:)

In this project you will compare saving plans. For instance, you deposit $1000 in a savings account and the following options are given:

a. 6.2% annual interest rate, compounded annually
b. 6.1% annual interest rate, compounded quarterly
c. 6.0% annual interest rate, compounded continuously


1) For each option, write a function that gives the balance as a function of the time t (in years).

2) Find the balance for the three options after 25, 50, 75 and 100 years. Is the option that yields the greatest balance after 25 years the same option that yields the greatest balances after 50, 75, 100 years? Explain!

3) Use a graphing utility to graph all three functions in the same viewing window. Can you find a viewing window that distinguishes among the graphs of the three functions? If so, describe the viewing window.

4) The effective yield of a savings plan is the percent increase in the balance after 1 year. Find the effective yields for three options listed above. How can the effective yield be used to decide which option is best?
 
Hello, mexx!

I'll get you started on these . . .

But if I assume you are familiar with the necessary formulas.
\(\displaystyle \;\;\)then exactly where is your difficulty?

Compound Interest: \(\displaystyle \L\:A\;=\;P\left(1\,+\,\frac{i}{n}\right)^{nt}\)

Compounded continuously: \(\displaystyle \L\,A\;=\;Pe^{it}\)

where: \(\displaystyle \,P\) = principal, \(\displaystyle i\) = annual interest rate, \(\displaystyle t\) = number of years,
. . . . . . . \(\displaystyle n\) = number of compounding periods per year.

You deposit $1000 in a savings account and the following options are given:

A) 6.2% annual interest rate, compounded annually
B) 6.1% annual interest rate, compounded quarterly
C) 6.0% annual interest rate, compounded continuously

1) For each option, write a function that gives the balance as a function of the time t (in years).
Click to expand...
Option A: \(\displaystyle \,P\,=\,\$1000,\;i\,=\.6.2\%\,=\,0.062,\;n\,=\,1\)

. . . . . . . . .\(\displaystyle A\;=\:1000(1\,+\,0.062)^t\)


Option B: \(\displaystyle \,P\,=\,\$1000,\;i\,=\,6.1\%\,=\,0.061,\;n\,=\,4\)

. . . . . . . . \(\displaystyle B\;=\;1000\left(1\,+\,\frac{0.061}{4}\right)^{4t}\)


Option C: \(\displaystyle \,P\,=\,\$1000,\;i\,=\,6%\,=\,0.06\)

. . . . . . . . \(\displaystyle C\;=\;1000e^{0.06t}\)


2) Find the balance for the three options after 25, 50, 75 and 100 years.
Is the option that yields the greatest balance after 25 years
the same option that yields the greatest balances after 50, 75, 100 years? Explain!
Click to expand...
I'll let you crank out the numbers . . .

It turns out that Option B always yields the greatest balance . . . here's why:

Option A: \(\displaystyle \,A\;=\;1000(1.062)^t\)

Option B: \(\displaystyle \,B\;=\;1000\left(1\,+\,\frac{0.061}{4}\right)^{4t}\;=\;1000\left[\left(1\,+\,\frac{0.061}{4}\right)^4 \right]^t \;=\;1000(1.062409615)^t\)

Option C: \(\displaystyle \,C\;=\;1000e^{0.6t}\;=\;1000\left(e^{0.6}\right)^t\;=\;1000(1.061836547)^t\)

We have three similar exponential functions, but Option B has the largest base
\(\displaystyle \;\;\). . . hence, it will always 'grow' faster.
 
could anybody give me a hint how to go about number 3 and 4?
 
3) For information on how to use your graphing calculator, please consult your owner's manual.

4) You've been given complete worked solutions for the hard half of the exercise. What have you done on this remaining bit?

Please reply with all your work and reasoning. Thank you.

Eliz.
 
#3, sorry i consulted the manual.. but i just don't understand how to tip it in the calculater.

for example:
a(1+(rn/n))^(n*t)

or in this case: 1000 (1+ (0.062/1)^(1*25)

then in the calculator:
Y1= ????
 
mexx said:
or in this case: 1000 (1+ (0.062/1)^(1*25)
Click to expand...
Be CAREFUL to apply brackets properly!

You need another one, at the end (also another * sign):
1000 *(1+ (0.062/1)^(1*25) )

And WHY are you entering it that way?
All you need is 1000*(1.062)^25
 
yes thanks, i know that i can enter it that way 1000*(1.062)^25.
but in the essay i have to write it the other way - don't wonder, my teacher insists on that way..

but how to enter the function in the calculator? still my question :?
 
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