The formula $A =19.6 e ^{0.0137}$ models the population of a US state, A, in millions, t years after 2000. Determine algebraically when the population was predicted to reach 25.9 million.
The population of the state will reach 25.9 million in the year $\square$
(Round up to the nearest year.)
Answer (2)
Substitute the given population into the formula: 25.9 = 19.6 e 0.0137 t .
Solve for t by dividing and taking the natural logarithm: t = 0.0137 l n ( 19.6 25.9 ) .
Calculate t ≈ 20.34 and round up to the nearest whole number: t = 21 .
Add t to the year 2000 to find the year the population reaches 25.9 million: 2000 + 21 = 2021 . The population of the state will reach 25.9 million in the year 2021 .
Explanation
Understanding the Problem We are given the formula A = 19.6 e 0.0137 t which models the population of a US state, A, in millions, t years after 2000. We want to find the year when the population reaches 25.9 million.
Setting up the Equation First, we substitute A = 25.9 into the formula: 25.9 = 19.6 e 0.0137 t Now, we need to solve for t .
Isolating the Exponential Term Divide both sides of the equation by 19.6: 19.6 25.9 = e 0.0137 t
Applying the Natural Logarithm Take the natural logarithm of both sides: l n ( 19.6 25.9 ) = l n ( e 0.0137 t ) l n ( 19.6 25.9 ) = 0.0137 t
Solving for t Now, solve for t by dividing both sides by 0.0137: t = 0.0137 l n ( 19.6 25.9 )
Calculating t Using a calculator, we find: t ≈ 0.0137 l n ( 1.3214 ) ≈ 0.0137 0.2785 ≈ 20.34
Rounding Up Since we need to round up to the nearest year, we have t = 21 years.
Finding the Year Finally, we add this to the year 2000 to find the year when the population is predicted to reach 25.9 million: 2000 + 21 = 2021
Examples
Population growth models are used in urban planning to predict future population sizes. This helps in resource allocation, infrastructure development, and policy making. For instance, knowing when a city's population will reach a certain threshold allows planners to prepare for increased demand on schools, hospitals, and transportation systems. These models also assist in understanding demographic trends and their impact on social services and economic growth.
The population of the state is predicted to reach 25.9 million in the year 2021. This was determined by substituting the value into the population model, isolating the exponential term, and calculating the year based on the growth function. Finally, we rounded the calculated time to the nearest whole year.
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