To determine how much more money Savannah would have in her account than Lincoln after 20 years, we need to use the formulas for compound interest. Lincoln's interest is compounded quarterly, while Savannah's interest is compounded continuously.
### Lincoln's Investment:
1. Initial Principal (P): \[tex]$6,500
2. Annual Interest Rate (r): 3.75% or \( \frac{3 + \frac{3}{4}}{100} = 0.0375 \)
3. Number of times interest applied per year (n): 4 (quarterly)
4. Time (t): 20 years
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Plugging in the values:
\[ A_{Lincoln} = 6500 \left(1 + \frac{0.0375}{4}\right)^{4 \times 20} \]
This simplifies to:
\[ A_{Lincoln} \approx 1.9640953388837747 \times 10^{23} \]
### Savannah's Investment:
1. Initial Principal (P): \$[/tex]6,500
2. Annual Interest Rate (r): 4.125% or [tex]\( \frac{4 + \frac{1}{8}}{100} = 0.04125 \)[/tex]
3. Time (t): 20 years
The formula for continuous compound interest is:
[tex]\[ A = Pe^{rt} \][/tex]
Plugging in the values:
[tex]\[ A_{Savannah} = 6500 e^{0.04125 \times 20} \][/tex]
This simplifies to:
[tex]\[ A_{Savannah} \approx 3.692574540092576 \times 10^{38} \][/tex]
### Difference in Amounts:
To find out how much more money Savannah would have than Lincoln:
[tex]\[ \text{Difference} = A_{Savannah} - A_{Lincoln} \][/tex]
[tex]\[ \text{Difference} \approx 3.692574540092576 \times 10^{38} - 1.9640953388837747 \times 10^{23} \][/tex]
The computed difference in their amounts is:
[tex]\[ \text{Difference} \approx 369257454009257347390927822749421273088 \][/tex]
Therefore, to the nearest dollar, Savannah would have \$369,257,454,009,257,347,390,927,822,749,421,273,088 more than Lincoln after 20 years.
