Certainly! We are given that:
- [tex]\( P = \$1900.00 \)[/tex] (the initial amount)
- [tex]\( r = 0.04 \)[/tex] (the annual rate of interest, which is 4%)
- [tex]\( n = 2 \)[/tex] (because the interest is compounded semi-annually)
- We want to find [tex]\( t \)[/tex] (the time in years for the investment to double)
- [tex]\( A = 2 \times P = 2 \times 1900 = \$3800 \)[/tex]
We need to solve for [tex]\( t \)[/tex] using the formula:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
1. First, let's isolate the term containing [tex]\( t \)[/tex]:
[tex]\[ \frac{A}{P} = \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
2. Substitute the known values:
[tex]\[ \frac{3800}{1900} = \left( 1 + \frac{0.04}{2} \right)^{2t} \][/tex]
3. Simplify the fraction:
[tex]\[ 2 = \left( 1 + 0.02 \right)^{2t} \][/tex]
[tex]\[ 2 = 1.02^{2t} \][/tex]
4. Take the natural logarithm of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \ln(2) = \ln(1.02^{2t}) \][/tex]
5. Using the properties of logarithms:
[tex]\[ \ln(2) = 2t \cdot \ln(1.02) \][/tex]
6. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(2)}{2 \cdot \ln(1.02)} \][/tex]
7. Substitute the known values of logarithms:
[tex]\[ t = \frac{0.6931471805599453}{2 \cdot 0.01980262729617973} \][/tex]
8. Simplify the value:
[tex]\[ t = \frac{0.6931471805599453}{0.03960525459235946} \][/tex]
[tex]\[ t \approx 17.501 \][/tex]
So, it will take approximately [tex]\( 17.501 \)[/tex] years for the investment to double when compounded semi-annually at [tex]\( 4\% \)[/tex].
Therefore, the time it will take is [tex]\( \boxed{17.501} \)[/tex] years.
