91Ó°ÊÓ

We are given that the original amount doubles in two years, therefore A = 2 * P, and t = 2. Using the continuous compound interest formula: \(2P = P * e^{2r}\) Now, we need to solve for r. Step 1: Divide by P on both sides of the equation: \(\frac{2P}{P} = \frac{P * e^{2r}}{P}\) Simplifying, we get: \(2 = e^{2r}\) Step 2: Use the natural logarithm function (ln) to eliminate e from the equation: \(ln(2) = ln(e^{2r})\) Step 3: Use the property of logarithms that allows us to bring the exponent down: \(ln(2) = 2r * ln(e)\) Since \(ln(e) = 1\), we have: \(ln(2) = 2r\) Step 4: Divide by 2 to solve for r: \(r = \frac{ln(2)}{2}\) The annual interest rate is r ≈ 0.3466, or approximately 34.66%.

We are given that the original amount increases by 50% in six months, so A = 1.5 * P, t = 0.5 (since six months is equal to 0.5 years). We will use the continuously compounded interest formula again: \(1.5P = P * e^{0.5r}\) Now we need to solve for the time it takes for the original amount to double, keeping in mind that we found the interest rate in part (a). Step 1: Divide by P on both sides of the equation: \(\frac{1.5P}{P} = \frac{P * e^{0.5r}}{P}\) Simplifying, we get: \(1.5 = e^{0.5r}\) Step 2: Substitute the annual interest rate found in part (a): \(1.5 = e^{0.5 * 0.3466}\) Solving it, we have: \(1.5 = e^{0.1733}\) Step 3: Use the natural logarithm function (ln) to eliminate e from the equation: \(ln(1.5) = ln(e^{0.1733})\) Step 4: Use the property of logarithms that allows us to bring the exponent down: \(ln(1.5) = 0.1733 * ln(e)\) Since \(ln(e) = 1\), we have: \(ln(1.5) = 0.1733\) Step 5: We know that the original amount doubles when A = 2*P: \(2P = P * e^{rt}\) Dividing by P, we get: \(2 = e^{rt}\) Step 6: We can replace r with the value found in part (a): \(2 = e^{0.3466t}\) Step 7: Use the natural logarithm function (ln) to eliminate e from the equation: \(ln(2) = ln(e^{0.3466t})\) Step 8: Use the property of logarithms that allows us to bring the exponent down: \(ln(2) = 0.3466t * ln(e)\) Since \(ln(e) = 1\), we have: \(ln(2) = 0.3466t\) Step 9: Divide by 0.3466 to solve for t: \(t = \frac{ln(2)}{0.3466}\) The time t ≈ 2 years. It takes approximately 2 years for the original amount to double when it increases by 50% in six months.