To determine how much money Eli will have in his account after four years with continuous compounding, we can use the formula:
\[ P(t) = P_0 e^{rt} \]
where:
- \( P(t) \) is the amount of money accumulated after time \( t \),
- \( P_0 \) is the principal amount (the initial amount of money),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
In this case, \( P_0 = 10,000 \), \( r = 0.05 \), and \( t = 4 \).
Plugging in these values, we have:
\[ P(4) = 10,000 e^{0.05 \times 4} \]
Calculating \( 0.05 \times 4 \):
\[ 0.05 \times 4 = 0.20 \]
Now plug this value into the equation:
\[ P(4) = 10,000 e^{0.20} \]
Now we need to calculate \( e^{0.20} \):
\[ e^{0.20} \approx 1.2214 \]
Now we can find \( P(4) \):
\[ P(4) = 10,000 \times 1.2214 \approx 12,214 \]
Thus, the amount of money in Eli's account after four years will be approximately:
\[ \boxed{12,214.00} \]
1 answer