Annual Interest Rate related question

HW1-3 (10\%). You deposit an amount K on an account with annual interest rate r0. (a) Use natural logarithms to find the time in years to double your capital as a function T of r. Hint: Write down the time value of money equation that you need to solve for T. You may use Wolfram Alpha to help you figure out the solution. Remember to show all your steps! (b) Use a computer to graph T(r) found in (a), together with the function r.72 for 1%r15%. What do you notice? (c) Based on your result in question (a), what interest rate do you need to double your capital in 10 years? (d) Let c(r)=ln(1+r). Calculate drdc as a function of r, then compute the value of drdc when r=0, and then use the approximation drdcrc(r)c(0) for r close to zero to argue that ln(1+r)r. (e) Use your results in questions (a) and (d) and the value ln2.693 to identify a rule of thumb (approximation) for T(r) when the interest rate r is low. How well does your rule of thumb agree with the rule of 72 described in the textbook $9.4.3 p. 239