The growth of the population of Los Angeles, California, for...

The growth of the population of Los Angeles, California, for the years 1992 through 1996 can be approximated by the equation $$P=10,130(1.005)^{\prime}$$ where $t=0$ corresponds to January 1,1992 and $P$ is in thousands. a. Assuming this growth rate continues, what will be the population of Los Angeles on January 1 in the year $2004 ?$ b. In what year will the population of Los Angeles first exceed $13,000,000 ?$

The growth of the population of Los Angeles, California, for the years 1992 through 1996 can be approximated by the equation
$$P=10,130(1.005)^{\prime}$$
where $t=0$ corresponds to January 1,1992 and $P$ is in thousands.
a. Assuming this growth rate continues, what will be the population of Los Angeles on January 1 in the year $2004 ?$
b. In what year will the population of Los Angeles first exceed $13,000,000 ?$

Key Concepts

Exponential Growth

Exponential growth describes processes in which the rate of increase of a quantity is proportional to its current value, leading to growth by constant multiplication over equal time periods. This concept is central to modeling phenomena like population growth, where small, consistent increases compound over time.

Population Modeling

Population modeling involves creating mathematical representations to predict changes in populations over time. Using exponential models allows one to estimate future population sizes given a constant growth rate, which is useful for planning and analysis in demography.

Exponential Equations

Exponential equations are mathematical statements where variables appear as exponents. These equations are pivotal in representing scenarios with compounded growth or decay, and solving them often involves isolating the exponent to determine time or rate values.

Logarithms and Their Use

Logarithms are the inverse of exponentiation and offer a way to solve equations in which the variable is an exponent. By applying logarithms to both sides of an exponential equation, one can algebraically extract the variable and find its value, a key step in determining when a population reaches a specific size.

Transcript

00:01 So what's important here is to realize what the context is talking about.

00:06 So i know that p represents the population in thousands of los angeles, california.

00:12 And t represents the years between 1992 and 1996.

00:18 So that means when t is zero, this is the year in 1992.

00:21 This will be important to remember as we solve our problem.

00:25 So first thing is let's say like this growth rate continues, right? i'm still growing exponentially because my multiplier is greater than one, and i start at 10 ,130 ,000, 10 ,130 million, 13 million people.

00:40 Let's say i want to go to the year 2004.

00:44 What will t equal? well, remember, i started at 1992, so i'm going to do 2004 minus 1992, how many years have passed since then.

00:54 This will be approximately 12 years since then.

00:58 So i'm going to plug in 12 into 20.

01:01 This equation and i'm just going to evaluate it in my calculator and what i'm going to get is approximately 10 ,754 .7 so that will round up to 755 and this is in thousands so i'm going to add another zero three zeros here.

01:31 Okay remember because p's in thousands.

01:34 Next let's say i wanted to figure out what year, okay, so now i'm solving for t, what year will the population first exceed 13 million? okay, remember that p is in thousands.

01:49 So when i plug in p, i just need to ignore the first three zeros...

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