To find the total amount of the investment after 32 years, given that the principal of $1,500 increases 1.5 times every 10 years, we can follow this step-by-step process.
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Determine the number of 10-year intervals in 32 years:
\[ \text{Number of intervals} = \frac{32}{10} = 3.2 \] This means we have 3 complete intervals of 10 years and an additional 2 years. -
Calculate the total growth after 30 years (3 intervals):
Since the investment increases 1.5 times every 10 years, after 30 years the amount is: \[ A = P \times (1.5)^n \] where \( P \) is the principal amount ($1,500), and \( n \) is the number of intervals (3). \[ A = 1500 \times (1.5)^3 \] Calculate \( (1.5)^3 \): \[ (1.5)^3 = 1.5 \times 1.5 \times 1.5 = 3.375 \] Now calculate the amount after 30 years: \[ A = 1500 \times 3.375 = 5062.5 \] -
Determine the growth for the additional 2 years:
The growth rate per year can be approximated. Since it increases by a factor of 1.5 every 10 years, we can find the annual growth factor: \[ r = (1.5)^{\frac{1}{10}} \] Using a calculator, we find: \[ r \approx 1.0414 \] Now, calculate the amount for these additional 2 years: \[ A_{2 \text{ years}} = A_{30 \text{ years}} \times (1.0414)^2 \] Calculate \( (1.0414)^2 \): \[ (1.0414)^2 \approx 1.086 \] Now update the total amount: \[ A_{32 \text{ years}} = 5062.5 \times 1.086 \approx 5492.85 \]
Thus, the estimated total amount of the account after 32 years is approximately $5,492.85.
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