Final answer:
To find the annual growth rate leading to a factor-of-ten increase over a century, one can use the natural logarithm of 10, which equates to a 2.3% growth rate for continuous growth. The relationship between the growth rate and the growth period can be expressed using formulas that take into account the present value, future value, the growth rate, and the time in years.
Step-by-step explanation:
The question pertains to finding the annual growth rate that would result in the investment increasing by a factor of ten over a century. Using a given equation often labelled as Eq. 1.5 or similar, we are looking for an annual growth rate, often denoted as 'g', which leads to exponential growth of the initial value of an investment.
According to the provided information, a 2.3% annual growth rate would yield a tenfold increase, i.e., a factor-of-ten growth in 100 years. This is derived from the natural logarithm of 10 being approximately 2.3, assuming continuous growth (as formalized in the provided equations). The relationship between the growth rate and the time it takes to grow by a certain factor can be illustrated through the formula Future Value = Present Value x (1 + g)^n, where 'g' is the growth rate and 'n' is the number of periods (years).
For example, if an investment had an annual growth rate of 7% (b = 1.07), it would take about 102 years to increase by a factor of 1,000, shown by the formula n = ln M/ln 1.07, where 'n' is the number of years and 'M' is the growth factor.