Problem 48 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The "Rule of \(70^{\text {"' }}\) says that if a quantity grows exponentially at a rate of \(r \%\) per unit of time, then its doubling time is usually about \(70 / r .\) This is merely a rule of thumb. Now we will determine how accurate an estimate this is and for what values of \(r\) it should be applied. Suppose that a quantity \(Q\) grows exponentially at \(r \%\) per unit of time \(t .\) Thus, \(Q(t)=Q_{0}\left(1+\frac{r}{100}\right)^{t}\) (a) Let \(D(r)\) be the doubling time of \(Q\) as a function of \(r .\) Find an equation for \(D(r)\). (b) On your graphing calculator, graph \(D(r)\) and \(70 / r .\) Take note of the values of \(r\) for which the latter is a good approximation of the former.

Step-by-Step Solution

Answer

The equation for the doubling time D(r) as a function of r is \( D(r) = \frac{\log(2)}{\log\left(1+\frac{r}{100}\right)} \). Upon graphing and comparing this function with the function 70/r, it can be observed where and how accurately the Rule of 70 estimates the doubling time for different rates of growth r.

Step 1: Find the Equation for Doubling Time

Start with the equation for exponential growth: \( Q(t)=Q_{0}\left(1+\frac{r}{100}\right)^{t} \). We need to find the doubling time, so we set \( Q(t) = 2Q_{0} \). This yields: \( 2Q_{0}=Q_{0}\left(1+\frac{r}{100}\right)^{t} \). After canceling out \( Q_{0} \) on both sides, we get: \( 2 = \left(1+\frac{r}{100}\right)^{t} \). To solve for t, apply the logarithm base 10 on both sides: \( \log(2) = t \log\left(1+\frac{r}{100}\right) \). Divide by \( \log\left(1+\frac{r}{100}\right) \) to solve for t, our doubling time D(r) as a function of r. Hence the equation is \( D(r) = \frac{\log(2)}{\log\left(1+\frac{r}{100}\right)} \).

Step 2: Compare The Real Doubling Time with The Estimated Doubling Time

Graph the function \( D(r) = \frac{\log(2)}{\log\left(1+\frac{r}{100}\right)} \) obtained in step 1 and compare it to the function 70/r, which is the estimate provided by the Rule of 70. Note the values of \( r \) for which the latter provides a good approximation to the former. The good approximations occur when these two curves intersect each other.

Key Concepts

Doubling Time

Doubling time is an important concept when we look at exponential growth. It tells us how long it takes for a quantity to double in size given a constant rate of growth. In the context of the given exercise, the doubling time can be calculated using the formula derived from setting the exponential growth equation equal to double the initial amount, or \( Q(t) = 2Q_0 \).
To find the exact doubling time, \( D(r) \), when a quantity grows at a rate \( r \% \), we solve the equation \( 2 = \left(1 + \frac{r}{100}\right)^t \). By applying logarithms, we obtain:

• Take logarithms on both sides: \( \log(2) = t \cdot \log\left(1 + \frac{r}{100}\right) \)
• Solve for \( t \): \( t = \frac{\log(2)}{\log\left(1 + \frac{r}{100}\right)} \)

This equation allows us to find the exact doubling time for any continuous exponential growth rate, providing us with an essential tool for understanding how rapidly growth is occurring or predicting future values.

Rule of 70

The Rule of 70 is a simple, widely used method for estimating the doubling time of an exponentially growing quantity. The idea is straightforward: if you know the annual growth rate \( r \% \), you can quickly approximate the time it takes for a quantity to double by using the formula \( \frac{70}{r} \). This rule serves as a practical shortcut because it bypasses more complex calculations.
However, it's important to remember that the Rule of 70 is not exact. It's an estimation that works well for moderate growth rates. The reason why 70 is used instead of another number is due to the approximation of natural logarithms used in the concept. The Rule of 70 is most accurate when the growth rates are not too large or too small, typically working best for growth rates between 0\% to 10\%. Outside of this range, the estimation may become less accurate, so other methods, including exact formulas, could be more appropriate.
Overall, the Rule of 70 provides a quick way to judge the doubling time without any detailed calculations, making it useful for business projections, finances, and population studies.

Logarithms

Logarithms are the mathematical concept that helps us simplify complex exponential calculations, like finding the doubling time. A logarithm tells us how many times we need to multiply a base number to get another number.
For instance, logarithms are particularly powerful in solving equations where the variable is an exponent, such as in exponential growth functions. In our exercise, to find doubling time, we transformed the equation \( 2 = \left(1 + \frac{r}{100}\right)^t \) by applying a logarithm:

• Taking \( \log_{10} \) on both sides leads to \( \log(2) = t \cdot \log\left(1 + \frac{r}{100}\right) \)
• Isolating \( t \) gives us \( t = \frac{\log(2)}{\log\left(1 + \frac{r}{100}\right)} \)

This logarithmic transformation helps us solve for \( t \), demonstrating the value of logarithms in breaking down equations that involve growth rates and time. Using logarithms simplifies our calculations, allowing us to manage exponential functions with ease and accuracy.