The population (in millions) of a state t years after 1970...

The population (in millions) of a state $t$ years after 1970 is given by the graph of the exponential function $y=P(t)$ with growth constant. 025 in Fig. 6. [In parts (c) and (d) use the differential equation satisfied by $P(t) .]$ (a) What was the population in $1974 ?$ (b) When was the population 10 million? (c) How fast was the population growing in $1974 ?$ (d) When is the population growing at the rate of 275,000 people per year?

The population (in millions) of a state $t$ years after 1970 is given by the graph of the exponential function $y=P(t)$ with growth constant. 025 in Fig. 6. [In parts (c) and (d) use the differential equation satisfied by $P(t) .]$
(a) What was the population in $1974 ?$
(b) When was the population 10 million?
(c) How fast was the population growing in $1974 ?$
(d) When is the population growing at the rate of 275,000 people per year?

Key Concepts

Solving Exponential Equations

To determine specific time points or population values in exponential growth, one often needs to solve exponential equations. This process usually involves applying logarithms to both sides of an equation in order to isolate the time variable or other parameters, which is essential for finding when a population reaches a particular value.

Rate of Change Evaluation

Evaluating the rate of change of an exponential function involves calculating the derivative of the function with respect to time. This derivative, given by the product of the growth constant and the current population, allows one to determine how rapidly the population is growing at any particular time and to solve for times when the growth rate meets a specified value.

Growth Constant

The growth constant is the proportionality factor in an exponential growth function that quantifies the rate at which the quantity grows. It appears in the exponent of the exponential function and determines how quickly the population increases. A higher constant indicates a faster growth rate, while a lower constant indicates slower growth.

Exponential Growth Function

An exponential growth function models situations where the rate of change of a quantity is proportional to its current value. In the context of population dynamics, this means that as the population increases, the absolute number of individuals added per unit time increases as well, leading to a multiplicative, rather than additive, increase over time.

Differential Equation for Exponential Growth

In modeling exponential growth, differential equations provide a fundamental framework. The typical form is dP/dt = kP, where 'P' is the population, 't' is time, and 'k' is the growth constant. This equation mathematically encapsulates the idea that the instantaneous rate of change of the population is directly proportional to the current population size.

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