a principal amount of $13,200 is placed in a savings account with an apr of 2.00% simple interest for 12…

a principal amount of $13,200 is placed in a savings account with an apr of 2.00% simple interest for 12 years. the same principal amount is placed in a savings account with an apr of 3.09% compounded continuously for 9 years. part a: how much interest is earned in the simple - interest account? show all necessary steps. (3 points) part b: how much interest is earned in the continuously compounded interest account? show all necessary steps. (3 points) part c: if the continuously compounded interest account was replaced with a monthly compounding account, what is the difference in the total account balances for the two compounded interest accounts? show all necessary steps. (4 points)

a principal amount of $13,200 is placed in a savings account with an apr of 2.00% simple interest for 12 years. the same principal amount is placed in a savings account with an apr of 3.09% compounded continuously for 9 years. part a: how much interest is earned in the simple - interest account? show all necessary steps. (3 points) part b: how much interest is earned in the continuously compounded interest account? show all necessary steps. (3 points) part c: if the continuously compounded interest account was replaced with a monthly compounding account, what is the difference in the total account balances for the two compounded interest accounts? show all necessary steps. (4 points)

Answer

Explanation:

Part A:

Step1: Recall simple - interest formula

The simple - interest formula is $I = Prt$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), and $t$ is the time in years. Given $P=$13200$, $r = 0.02$ (since $2.00%=0.02$), and $t = 12$ years. $I=Prt=13200\times0.02\times12$

Step2: Calculate the interest

$I = 13200\times0.02\times12=13200\times0.24 = 3168$

Part B:

Step1: Recall continuous - compounding formula

The formula for continuous compounding is $A=Pe^{rt}$, and the interest earned $I=A - P=P(e^{rt}-1)$. Given $P = 13200$, $r=0.0309$ (since $3.09% = 0.0309$), and $t = 9$ years. $I=13200\times(e^{0.0309\times9}-1)$

Step2: Calculate the exponent

First, calculate $0.0309\times9 = 0.2781$. Then, find $e^{0.2781}\approx1.3207$. $I=13200\times(1.3207 - 1)=13200\times0.3207=4233.24$

Part C:

Step1: Recall monthly - compounding formula

The formula for compound - interest is $A=P(1+\frac{r}{n})^{nt}$, where $n = 12$ (monthly compounding), $P = 13200$, $r=0.0309$, and $t = 9$ years. $A_{monthly}=13200\times(1+\frac{0.0309}{12})^{12\times9}=13200\times(1 + 0.002575)^{108}$ $(1 + 0.002575)^{108}\approx1.3201$ $A_{monthly}=13200\times1.3201=17425.32$ The amount in the continuous - compounding account (from Part B) is $A_{continuous}=13200 + 4233.24=17433.24$

Step2: Calculate the difference

$|A_{continuous}-A_{monthly}|=|17433.24 - 17425.32| = 7.92$

Answer:

Part A: $$3168$ Part B: $$4233.24$ Part C: $$7.92$