32) Rules of 70 and 72: The rules state that it takes about 70/i of 72/ i years for money to double at i percent, compounded continuously, using whichever of 70 or 72 is easier to divide by i. a) Show that it takes t=ln 2/r years for money to double if it is invested at annual interest rate r (in decimal form) compounded continuously. b) Graph the functions y_1=ln2 / r, y_2=70/100r, y_3=72/100r 0<=r<=0.1 and 0<=y<=100 viewing window. c) Explain why these two rules of thumb for mental computations are reasonable.

**(a) Derivation** To find out how long it takes for money to double with continuous compounding, we use the formula: \[ N(t) = N_0 e^{r t} \] Setting \(N(t) = 2N_0\), we get: \[ 2 = e^{r t} \implies t = \frac{\ln 2}{r} \] So, the time \(t\) it takes for money to double is \(\frac{\ln 2}{r}\) years. **(b) Graphing the Functions** We need to graph the following functions over the interval \(0 \leq r \leq 0.1\) and \(0 \leq y \leq 100\): 1. \(y_1 = \frac{\ln 2}{r}\) 2. \(y_2 = \frac{70}{100r}\) 3. \(y_3 = \frac{72}{100r}\) These functions represent the exact doubling time and the Rule of 70 and Rule of 72 estimates. **(c) Explanation of the Rules** The exact doubling time is \(\frac{\ln 2}{r}\). The Rules of 70 and 72 estimate this as \(\frac{70}{100r}\) and \(\frac{72}{100r}\) respectively. These rules are reasonable because for typical interest rates (3% to 10%), the estimates are close to the exact value, making them useful for quick mental calculations.