32) Rules of 70 and 72 . The rules state that it takes about $$ 70 / \mathrm{j} $$ of $$ 72 / $$ 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 rater (in decimal form) compounded continuously. $$ \begin{array}{l} N(4)=N_{e} e^{2 t} \ 2 v_{3}{ }^{2} M_{4} e^{r t} \ a=e^{+t}=\ln 2=r t \quad t=\ln (2) \end{array} $$ b) Graph the functions $$ y_{1}=\frac{2 m 2}{r} \quad y_{2}=\frac{70}{100 r} \quad y_{3}=\frac{72}{100 r} $$ in the $$ [0,0.1] $$ by $$ [0,100] $$ viewing window (c) Explain why these two rules of thumb for mental computations are reasonable.

Question

UpStudy AI · Accepted Answer

**(a) Derivation**

We start from the continuous compounding formula for an initial amount $$N_0$$:

$$
N(t)=N_0 e^{r t},
$$

where $$r$$ is the annual interest rate written in decimal form. To find the doubling time we set

$$
N(t)=2N_0.
$$

Thus,

$$
2N_0 = N_0 e^{r t}.
$$

Dividing both sides by $$N_0$$ (assuming $$N_0 \neq 0$$) gives

$$
2 = e^{r t}.
$$

Taking the natural logarithm of both sides, we have

$$
\ln 2 = r t.
$$

Solving for $$t$$,

$$
t=\frac{\ln 2}{r}.
$$

---

**(b) Graphing the Functions**

We are to graph the following functions over the viewing window $$[0,0.1]$$ for $$r$$ (the horizontal axis) and $$[0,100]$$ for $$y$$ (the vertical axis):

1. $$\displaystyle y_{1}=\frac{\ln 2}{r}$$.

2. $$\displaystyle y_{2}=\frac{70}{100 \, r}$$.  
   (This represents the Rule of 70 where the interest rate as a percentage is $$100r$$.)

3. $$\displaystyle y_{3}=\frac{72}{100 \, r}$$.  
   (This represents the Rule of 72.)

*A few notes on the graphs:*

- For very small values of $$r$$ (approaching 0), each function tends to infinity.  
- As $$r$$ increases within the interval $$[0,0.1]$$, the values of $$y_1$$, $$y_2$$, and $$y_3$$ decrease.  
- For example, at $$r=0.05$$ (which corresponds to 5% annual interest):

$$
  y_1=\frac{\ln 2}{0.05}\approx\frac{0.6931}{0.05}\approx13.86,
  $$
  
  $$
  y_2=\frac{70}{100\times0.05}=\frac{70}{5}=14,
  $$
  
  $$
  y_3=\frac{72}{100\times0.05}=\frac{72}{5}=14.4.
  $$

To graph these functions, plot each as a curve with $$r$$ on the horizontal axis and the corresponding $$y$$-value on the vertical axis. Notice that on the interval $$0**(a) Derivation** To find out how long it takes for money to double with continuous compounding, we use the formula: $$ 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.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions