To find out how long it will take Jasmine to grow her investment to $100,000, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (in this case, $100,000)
P = the initial principal (in this case, $73,000)
r = the annual interest rate (in this case, 7.2% or 0.072)
n = the number of times the interest is compounded per year (in this case, semiannually, so 2)
t = the time in years
Plugging in the given values, the equation becomes:
100,000 = 73,000(1 + 0.072/2)^(2t)
Now, we can solve for t. Let's simplify the equation further:
100,000 = 73,000(1.036)^2t
Divide both sides of the equation by 73,000:
1.3699 = (1.036)^2t
Next, take the natural logarithm (ln) of both sides of the equation:
ln(1.3699) = ln[(1.036)^2t]
Using the logarithm property ln(a^b) = b * ln(a):
ln(1.3699) = 2t * ln(1.036)
Now, divide both sides of the equation by 2 * ln(1.036):
t = ln(1.3699) / (2 * ln(1.036))
Using a calculator, we can find that ln(1.3699) ≈ 0.3156 and ln(1.036) ≈ 0.0351:
t = 0.3156 / (2 * 0.0351)
Simplifying further:
t ≈ 4.51
Therefore, it will take approximately 4.51 years for Jasmine to grow her investment to $100,000.