Chapter 8.2, Problem 39ES

(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 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 $1.08\%$ per year the world population would double every 64 years. This result agrees with the Rule of 70, since $70/1.08\approx \mathrm{64.8.}$ 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.