The population of a town grew from 20,000 to 28,000. The continuous growth rate is [tex]$15 \%$[/tex]. The equation [tex]$20,000 e^{0.15 t}=28,000$[/tex] represents the situation, where [tex]$t$[/tex] is the number of years the population has been growing. About how many years has the population of the town been growing? Use a calculator and round your answer to the nearest whole number.
A. 2 years
B. 9 years
C. 17 years
D. 22 years
Answer (2)
Divide both sides of the equation by 20,000: e 0.15 t = 1.4 .
Take the natural logarithm of both sides: 0.15 t = l n ( 1.4 ) .
Solve for t: t = 0.15 l n ( 1.4 ) .
Calculate and round t to the nearest whole number: t ≈ 2 .
Explanation
Understanding the Problem We are given the equation 20 , 000 e 0.15 t = 28 , 000 which models the population growth of a town, where t is the number of years the population has been growing. Our goal is to find the value of t , rounded to the nearest whole number.
Isolating the Exponential Term First, we divide both sides of the equation by 20,000 to isolate the exponential term: 20 , 000 20 , 000 e 0.15 t = 20 , 000 28 , 000 e 0.15 t = 1.4
Applying Natural Logarithm Next, we take the natural logarithm (ln) of both sides of the equation to eliminate the exponential: l n ( e 0.15 t ) = l n ( 1.4 ) Using the property of logarithms, l n ( e x ) = x , we get: 0.15 t = l n ( 1.4 )
Solving for t Now, we solve for t by dividing both sides by 0.15: t = 0.15 l n ( 1.4 )
Calculating t Using a calculator, we find the value of t :
t ≈ 2.243
Rounding to the Nearest Whole Number Finally, we round the value of t to the nearest whole number: t ≈ 2 Therefore, the population of the town has been growing for approximately 2 years.
Examples
Understanding exponential growth is crucial in many real-world scenarios. For instance, it helps in predicting the spread of diseases, calculating investment returns, and estimating resource depletion rates. In epidemiology, exponential growth models can forecast how quickly a virus might propagate through a population, informing public health strategies. Similarly, in finance, these models are used to project the growth of investments, aiding in financial planning. By grasping the principles of exponential growth, individuals and organizations can make more informed decisions in various aspects of life.
The population of the town has been growing for approximately 2 years. This is determined from the equation modeling its growth by isolating the exponential term, applying the natural logarithm, and solving for time. The answer that best fits the situation is option A: 2 years.
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