In the question, continuously should be annually.
Solution:
Applicable formula is;
A = P(1+r)^n
Where;
A = Total amount after 30 years = $9,110
P = Amount invested = $5,000
r = Annual interest rate in decimals
n = Number of years = 30
Substituting;
9110 = 5000(1+r)^30
9110/5000 = (1+r)^30
1.822 = (1+r)^30
Taking natural logs on both sides;
ln (1.822) = 30 ln (1+r)
0.5999 = 30 ln (1+r)
0.5999/30 = ln (1+r)
0.019998 = ln (1+r)
Taking exponents on both sides
e^0.019998 = 1+r
1.0202 = 1+r
r = 1.0202 -1 = 0.0202 =2.02%
Therefore, annual interest rate should be 2.02%.
