In 2007,1730 million people were citizens of India. Find how...

In $2007,1730$ million people were citizens of India. Find how long it will take India's population to reach a size of $2000 \mathrm{mil}$ lion (that is, 2 billion) if the population size is growing at a rate of $1.6 \%$ per year. Use $y=y_{0} e^{0.016 t}$ and round to the nearest tenth. (Source: U.S. Bureau of the Census, International Data Base)

Key Concepts

Logarithmic Manipulation in Solving Exponential Equations

Using logarithmic properties is key when one needs to solve for a variable in the exponent of an exponential function. This involves applying the logarithm to both sides of the equation, utilizing properties such as ln(e^(rt)) = rt, which facilitates the process of isolating and calculating the unknown variable, such as time in growth scenarios.

Natural Logarithms

Natural logarithms, particularly ln, are used to solve equations where the unknown variable appears as an exponent. In contexts like the growth model y = y?e^(rt), taking the natural logarithm of both sides allows one to isolate and solve for the time variable, providing a method to determine how long it takes for a quantity to reach a certain level.

Exponential Growth

Exponential growth describes a process where the rate of increase of a quantity is proportional to its current value. This concept is essential for modeling situations where change happens continuously over time, such as in population dynamics, where each increment contributes to a larger subsequent increase.

Continuous Growth Model

The continuous growth model is expressed as y = y?e^(rt), where y? represents the initial quantity, r the constant growth rate, and t the time elapsed. This model effectively demonstrates situations where growth occurs without interruption, making it crucial for understanding how populations or investments increase over time.

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