Problem 39 - Calculus Early Transcendentals

Question

(a) There is a trick, called the Rule of \(70,\) that can be used to get a quick estimate of the doubling time or half-life of an exponential model. According to to this rule, the doubling time or half-life is roughly 70 divided by the percentage growth or decay rate. For example, we showed in Example 5 that with a continued growth rate of \(1.33 \%\) per year the world population would double every 52 years. This result agrees with the Rule of \(70,\) since \(70 / 1.33 \approx 52.6 .\) Explain why this rule works. (b) Use the Rule of 70 to estimate the doubling time of a population that grows exponentially at a rate of \(1 \%\) per year. (c) Use the Rule of 70 to estimate the half-life of a population that decreases exponentially at a rate of \(3.5 \%\) per hour. (d) Use the Rule of 70 to estimate the growth rate that would be required for a population growing exponentially to double every 10 years.

Step-by-Step Solution

Answer

(b) 70 years, (c) 20 hours, (d) 7% growth rate.

Step 1: Understanding the Rule of 70

The Rule of 70 is a simple way to estimate the doubling time or half-life of a quantity that increases or decreases exponentially. You divide 70 by the annual (or any time period) percentage growth or decay rate.

Step 2: Explain Why the Rule of 70 Works

The doubling time formula for exponential growth is given by \( t = \frac{ ext{ln}(2)}{r} \). For small growth rates, \( ext{ln}(2) \approx 0.693 \) can be approximated by 0.7, which is how the Rule of 70 comes into play, leading to \( t \approx \frac{70}{r} \).

Step 3: Calculate Doubling Time at 1% Growth Rate

Using the Rule of 70: \( t = \frac{70}{1} = 70 \) years. Therefore, a population growing at 1% per year will double approximately every 70 years.

Step 4: Calculate Half-Life at 3.5% Decay Rate

Using the Rule of 70: \( t = \frac{70}{3.5} = 20 \) hours. The half-life of a population decreasing at a rate of 3.5% per hour is approximately 20 hours.

Step 5: Estimate Growth Rate for Doubling Every 10 Years

Using the Rule of 70, the equation is \( 70 = r \times 10 \). Solving for \( r \) gives: \( r = \frac{70}{10} = 7 \). Therefore, a growth rate of 7% per year would be required for a population to double every 10 years.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases by a fixed percentage over equal time intervals. This type of growth is characterized by its accelerating pace, where the amount grows larger with each time frame. In mathematical terms, the exponential growth formula is usually expressed as:

• \[ N(t) = N_0 imes e^{rt} \],

where:

• \( N(t) \) is the quantity at time \( t \),
• \( N_0 \) is the initial quantity,
• \( r \) is the growth rate, and
• \( t \) is the time elapsed.

Exponential growth is common in population studies, finance, and other fields. It’s important to note that for very small rates, an approximation can simplify calculations.For instance, when growth rates are small, the exponential function \( e^{rt} \) can be approximated easier, making the Rule of 70 helpful for quick estimates.

Doubling Time

Doubling time refers to the length of time it takes for a population or quantity to double in size or value at a constant growth rate. This concept is particularly useful when analyzing exponential growth patterns. Using the Rule of 70, doubling time can be easily estimated by dividing 70 by the percentage growth rate:

• \( t_{\text{double}} = \frac{70}{r} \).

"\( r \)" is the growth rate expressed as a percentage. This approximation arises from the natural logarithm properties, where for small rates, the factor \( \ln(2) \approx 0.7 \). Thus, a growth rate of 1% per year results in a doubling time of approximately 70 years according to this simplified calculation, making it a handy tool for quick predictions in real-world scenarios, such as population studies.

Half-Life

Half-life is a concept used to describe the time it takes for a quantity to decrease to half its original size. This is mainly a term used in contexts like radioactivity and pharmacokinetics but can extend to any exponentially decaying quantity. The Rule of 70 assists in estimating half-life by offering an easy formula, similar to calculating doubling time:

• \( t_{\text{half-life}} = \frac{70}{r} \),

where \( r \) is the percentage decay rate. An example is a decrease at a rate of 3.5% per hour, leading to a half-life of roughly 20 hours. The approximation makes using the Rule of 70 straightforward for quick estimates without dealing with logarithms directly, highlighting how certain mathematical properties of logarithms apply consistently across different types of growth and decay scenarios.

Growth Rate Calculation

Calculating the growth rate is essential to understand how fast a quantity is increasing in an exponential growth mode. The Rule of 70 also aids in deriving growth rates when you know the doubling time. Reworking the Rule of 70 formula gives you:

• \( r = \frac{70}{t_{\text{double}}} \).

Here, \( t_{\text{double}} \) is the time it takes for the quantity to double. For example, if a population doubles every 10 years, you can determine its annual growth rate to be approximately 7%. This is calculated by solving for \( r \) in the equation \( 70 = r \times 10 \). The Rule of 70 allows you to skip the complexity of logarithmic equations and quickly grab an estimate of both growth rate and time, making it a favored tool in finance and economics.