Solve each problem. The number of years, n, since two...

Solve each problem. The number of years, $n$, since two independently evolving languages split off from a common ancestral language is approximated by $$ n \approx-7600 \log r $$ where $r$ is the proportion of words from the ancestral language common to both languages. Find each of the following to the nearest year. (a) Find $n$ if $r=0.9$ (b) Find $n$ if $r=0.3$ (c) How many years have elapsed since the split if half of the words of the ancestral language are common to both languages?

Solve each problem.
The number of years, $n$, since two independently evolving languages split off from a common ancestral language is approximated by
$$
n \approx-7600 \log r
$$
where $r$ is the proportion of words from the ancestral language common to both languages. Find each of the following to the nearest year.
(a) Find $n$ if $r=0.9$
(b) Find $n$ if $r=0.3$
(c) How many years have elapsed since the split if half of the words of the ancestral language are common to both languages?

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Key Concepts

Rounding

Rounding is the process of approximating a number to a desired level of precision, such as the nearest whole number. It is important when final results must meet practical constraints, ensuring that calculations align with real-world measurements.

Substitution and Evaluation

Substitution involves replacing variables with specific numerical values in an equation to evaluate an outcome. This step is crucial when applying a general formula to particular cases in order to obtain meaningful estimates.

Logarithms

Logarithms are the inverse operation to exponentiation, transforming multiplicative relationships into additive ones. They are used to solve equations where the exponent is unknown, as well as to model natural processes like exponential decay or growth.

Mathematical Modeling

Mathematical modeling translates real-world phenomena into mathematical expressions. In contexts such as linguistics or biology, models can estimate growth, decay, or divergence times using functions that encapsulate the underlying processes.

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