The population of a town can be modeled by the regression equation y= 18,000(1.04). Which is the best prediction for the population in year 30?
A. 72,567
B. 68,298
C. 39,440
D. 58,381
Answer (2)
To find the population in year 30, we substitute t = 30 into the equation y = 18000 ( 1.04 ) 30 and calculate to find that y ≈ 58 , 381 . Thus, the best prediction for the population is option D: 58,381.
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Substitute the year t = 30 into the population model y = 18000 ( 1.04 ) t .
Calculate the population: y = 18000 ( 1.04 ) 30 .
Evaluate the expression: y ≈ 58381.16 .
The predicted population in year 30 is 58 , 381 .
Explanation
Understanding the Problem We are given the regression equation y = 18000 ( 1.04 ) t , where y represents the population and t represents the year. We want to find the population in year 30, so we need to substitute t = 30 into the equation.
Substituting the Value Substitute t = 30 into the equation: y = 18000 ( 1.04 ) 30 .
Calculating the Population Calculate the value of y : y = 18000 × ( 1.04 ) 30 = 18000 × 3.243397519 ≈ 58381.16 .
Finding the Best Prediction The best prediction for the population in year 30 is approximately 58,381.
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 example, a city planner can use this model to estimate the number of schools, hospitals, and roads needed in the future. Understanding exponential growth is crucial for managing resources effectively and ensuring sustainable development.
