Respuesta :
a)
Since the bank pays 5% each year, then the balance after one year will be 105% of the original balance.
To find the balance after one year, calculate what is 105% of 2000 equal to by multiplying 2000 times 105/100:
[tex]2000\times\frac{105}{100}=2000\times1.05=2100[/tex]The same process is repeated the next year, where the balance will be equal to 105% of 2100:
[tex]2100\times\frac{105}{100}=2100\times1.05=2205[/tex]And so on. To find the balance on the third year, multiply 2205 by 1.05:
[tex]2205\times1.05=2315.25[/tex]The balance on the fourth year will be:
[tex]2315.25\times1.05=2431.0125[/tex]And in the fifth year:
[tex]2431.0125\times1.05=2552.563125[/tex]Therefore, the table will include the following data:
[tex]\begin{matrix}\text{Year} & \text{Balance} \\ 1 & 2100 \\ 2 & 2205 \\ 3 & 2315.25 \\ 4 & 2431.0125 \\ 5 & 2252.563125 \\ \end{matrix}[/tex]B)
Since each year the balance gets multiplied by 1.05, then after t years the balance would increase by a factor of 1.05^t.
Since at year 0 the balance was $2000, then the equation that models the balance b after t years is:
[tex]b=2000\times1.05^t[/tex]C)
To find how many years it will take for the original deposit to double in value, set b=4000 and solve for t:
[tex]\begin{gathered} 4000=2000\times1.05^t \\ \Rightarrow\frac{4000}{2000}=1.05^t \\ \Rightarrow2=1.05^t \\ \Rightarrow t=\log _{1.05}(2) \\ \therefore t=14.2\approx14 \end{gathered}[/tex]It will take approximately 14 years for the balance to double.
D)
If the interest rate was 10%, then each year the balance would increase by a factor of 110/100, which is equal to 1.1.
Then, the model for the balance as a function of time would be:
[tex]b=2000\times1.1^t[/tex]And the time that it would take for the balance to double would be:
[tex]t=\log _{1.1}(2)=7.27\approx7[/tex]Then, it would take approximately 7 years for the balance to double if the interest rate was 10% instead of 5%.
