obrozenecy6

2021-12-30

Andy and Ena opened separate bank accounts that earn compound interest. Andy's account can be modeled by the function $f\left(x\right)=275{\left(1.06\right)}^{x}$. Ena's account can be modeled by the function $f\left(x\right)=115{\left(1.09\right)}^{x}$. Which statement about the functions is true?
F. The beginhing balances cannot be determined, but Andy's account earns a higher annual interest rate than Ena's account.
G. The beginning balances cannot be determined, but Ena's account earns a higher annual interest rate than Andy's account.
H. Andy's account has a higher beginning balance, but Ena's account earns a higher annual interest rate of 9%.
J. Andy's account has a higher beginning balance, but Ena's account earns a higher annual interest rate of 1.09%.

sonSnubsreose6v

Step 1
The amount(A) in the account after x years can be calculated as below, where P is the principal and interest rate is r% and the time of deposit is x.
$A=P{\left(1+\frac{r}{100}\right)}^{x}$
Step 2
Write Andy’s and Ena’s account function as above function and find the beginning balance and interest rates.
For Andy $F\left(x\right)=275{\left(1.06\right)}^{x}$
$F\left(x\right)=275{\left(1+\frac{6}{100}\right)}^{x}$
So, beginning balance $=275$
interest rate $=6\mathrm{%}$
For Era $F\left(x\right)=115{\left(1.09\right)}^{x}$
$F\left(x\right)=115{\left(1+\frac{9}{100}\right)}^{x}$
So, beginning balance $=115$
interest rate $=9\mathrm{%}$
Hence, Statement (H) is True.
Andy’s account has higher beginning balance but Ena’s account earns a higher annual interest rate of 9%.

encolatgehu

Step 1
Given,
Andy and Ana’s account modeled by the function $f\left(x\right)=275{\left(1.06\right)}^{x}$ and $f\left(x\right)=115{\left(1.09\right)}^{x}$.
As we know, $A=P{\left(1+\frac{r}{100}\right)}^{t}$ (i)
Where $A=$ Final Amount
$P=$ Initial Principal Value
$R=$ Rate
$T=$ time
Step 2
The beginning the balance can be calculated by substituting $x=0$ as shown below:
The beginning the balance of Andy $=f\left(0\right)=275{\left(1.06\right)}^{0}=275$
And the beginning the balance of Ana’s $=f\left(0\right)=115{\left(1.09\right)}^{0}=115$
So, the beginning balance of Andy is more than the beginning balance of Ana’s.
Let the interest rate of Andy and Ana's be
Now, rearranging the account function of Andy and Ana's as shown below:
$f\left(x\right)=275{\left(1.06\right)}^{x}=275{\left(1+0.06\right)}^{x}=275{\left(1+\frac{6}{100}\right)}^{x}$ (ii)
and $f\left(x\right)=115{\left(1.09\right)}^{x}=115{\left(1+0.09\right)}^{x}=115{\left(1+\frac{9}{100}\right)}^{x}$ (iii)
Now, comparing the equation (ii) and (iii) with respect to the equation (i), we get

Since, ${r}_{1}<{r}_{2}$ so the interest rate of Andy is less than the interest rate of Ana's
Step 3
Answer: The correct Option is H.

user_27qwe

The amount(A) in the account after x years can be calculated as below, where P is the principal and interest rate is r% and the time of deposit is x.
$A=P\left(1+\frac{r}{100}{\right)}^{x}$
Write Andy’s and Ena’s account function as above function and find the beginning balance and interest rates.
For Andy
$F\left(x\right)=275\left(1+\frac{6}{100}{\right)}^{x}$
=275
interest rate =6%
For Era $F\left(x\right)=115\left(1.09{\right)}^{x}$
$F\left(x\right)=115\left(1+\frac{9}{100}{\right)}^{x}$
=115
interest rate =9%
Hence, Statement (H) is True.
Andy’s account has higher beginning balance but Ena’s account earns a higher annual interest rate of 9%.

Do you have a similar question?