open - ended describe two account options into which you can deposit $1000 and earn compound interest. write…

open - ended describe two account options into which you can deposit $1000 and earn compound interest. write a function that represents the balance of each account after t years. which account would you rather use? explain your reasoning.

open - ended describe two account options into which you can deposit $1000 and earn compound interest. write a function that represents the balance of each account after t years. which account would you rather use? explain your reasoning.

Answer

Option 1: Savings Account (Quarterly Compounding)

Brief Explanations:

A common savings account might have an annual interest rate of ( r = 0.02 ) (2%) and compound quarterly (( n = 4 )). The compound - interest formula is ( A(t)=P\left(1 + \frac{r}{n}\right)^{nt} ), where ( P=$1000 ) (principal), ( r = 0.02 ), ( n = 4 ), and ( t ) is the number of years. Substituting the values, we get ( A_1(t)=1000\left(1+\frac{0.02}{4}\right)^{4t}=1000(1 + 0.005)^{4t}=1000(1.005)^{4t}).

Option 2: Certificate of Deposit (CD, Annual Compounding)

Brief Explanations:

A CD might have a higher annual interest rate, say ( r = 0.03 ) (3%) and compound annually (( n = 1 )). Using the compound - interest formula ( A(t)=P\left(1+\frac{r}{n}\right)^{nt} ) with ( P = 1000 ), ( r=0.03 ), ( n = 1 ): ( A_2(t)=1000\left(1+\frac{0.03}{1}\right)^{1\times t}=1000(1.03)^{t})

Comparison and Choice

Brief Explanations:

To compare the two accounts, we can analyze the growth rate. Let's consider the effective annual rate (EAR) for the savings account. The EAR formula is ( EAR=\left(1+\frac{r}{n}\right)^{n}-1 ). For the savings account, ( EAR=(1 + 0.005)^{4}-1\approx(1.005)^{4}-1\approx1.02015 - 1=0.02015) or 2.015%. For the CD, the EAR is 3%. Since the CD has a higher effective annual interest rate, over time, the balance in the CD will grow faster than the balance in the savings account. So, if the goal is to maximize the return on the $1000 deposit over time, the CD account (with ( A_2(t)=1000(1.03)^{t} )) is a better choice. However, savings accounts are more liquid (you can withdraw money more easily), while CDs often have penalties for early withdrawal. If liquidity is a major concern, the savings account might be preferred, but from a pure return - on - investment perspective, the CD is better.

Answer:

  1. Savings Account (Quarterly Compounding): Function is ( A_1(t)=1000(1.005)^{4t} ) (assuming 2% annual interest, compounded quarterly).
  2. CD (Annual Compounding): Function is ( A_2(t)=1000(1.03)^{t} ) (assuming 3% annual interest, compounded annually).
  3. Choice: If return on investment is the main concern, the CD account (( A_2(t)=1000(1.03)^{t} )) is better because it has a higher effective annual interest rate (3% vs. approximately 2.015% for the savings account) and will grow the $1000 deposit faster over time. If liquidity is more important, the savings account (( A_1(t)=1000(1.005)^{4t} )) may be preferred as it is more accessible.