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Chapter 4: Problem 20

About how many years does it take for \(\$ 300\) to become \(\$ 2,400\) when compounded continuously at \(5 \%\) per year?

Short Answer

Expert verified
It takes approximately \(32.27\) years for $\$ 300$ to grow to $\$ 2,400$ when compounded continuously at a 5% interest rate per year.

Step by step solution

01

Identify the given values

We are given: - Principal amount (P) is $300. - Final amount (A) is $2,400. - Interest rate (r) is 5% per year. Note that we need to convert the interest rate from percentage to decimal form by dividing by 100. So, r = 0.05. Our goal is to find the time, t, in years.
02

Plug the given values into the continuous compounding formula

We will use the formula \(A = Pe^{rt}\) and substitute the given values: \(2,400 = 300e^{(0.05)t}\)
03

Solve for t

To find t, follow these steps: 1. Divide both sides of the equation by 300: \(\frac{2,400}{300} = e^{(0.05)t}\) 2. Simplify the left side: \(8 = e^{(0.05)t}\) 3. Take the natural logarithm of both sides: \(\ln(8) = \ln\left(e^{(0.05)t}\right)\) 4. Use the logarithm property that states \(\ln(a^b) = b\ln(a)\): \(\ln(8) = (0.05)t\ln(e)\) 5. Since \(\ln(e) = 1\), the equation simplifies to: \(\ln(8) = (0.05)t\) 6. Now, divide both sides of the equation by 0.05: \(\frac{\ln(8)}{0.05} = t\) 7. Finally, calculate the value of t: \(t \approx 32.27\)
04

Interpret the result

It will take approximately 32.27 years for the initial amount of \(300 to grow to \)2,400 when compounded continuously at an interest rate of 5% per year.

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