Questions: An initial investment amount P, an annual interest rate r, and a time t are given. Find the future the doubling time T for the given interest rate. P= 450, r=2.35%, t=12 yr

An initial investment amount P, an annual interest rate r, and a time t are given. Find the future the doubling time T for the given interest rate. P= 450, r=2.35%, t=12 yr
Transcript text: An initial investment amount $P$, an annual interest rate $r$, and a time $t$ are given. Find the futur the doubling time T for the given interest rate. \[ P=\$ 450, r=2.35 \%, t=12 \mathrm{yr} \]
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Solution

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Solution Steps

To find the future value of an investment with compound interest, we use the formula \( A = P(1 + \frac{r}{100})^t \). To find the doubling time \( T \), we use the rule of 70, which is an approximation given by \( T \approx \frac{70}{r} \).

Step 1: Calculate Future Value

To find the future value \( A \) of the investment after \( t = 12 \) years, we use the formula for compound interest:

\[ A = P \left(1 + \frac{r}{100}\right)^t \]

Substituting the given values \( P = 450 \), \( r = 2.35 \), and \( t = 12 \):

\[ A = 450 \left(1 + \frac{2.35}{100}\right)^{12} \approx 594.6572 \]

Step 2: Calculate Doubling Time

To find the doubling time \( T \) for the given interest rate, we can use the rule of 70:

\[ T \approx \frac{70}{r} \]

Substituting \( r = 2.35 \):

\[ T \approx \frac{70}{2.35} \approx 29.7872 \]

Final Answer

The future value of the investment after 12 years is approximately \( A \approx 594.6572 \) and the doubling time is approximately \( T \approx 29.7872 \).

Thus, the final answers are: \[ \boxed{A \approx 594.6572} \] \[ \boxed{T \approx 29.7872} \]

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