The number of fish in a lake can be modeled by the exponential regression equation $y=14.08 \cdot 2.08^x$, where $x$ represents the year.
Which is the best prediction for the number of fish in year 6? Round your answer to the nearest whole number.
A. 81
B. 1140
C. 1758
D. 176
Answer (1)
Substitute x = 6 into the equation y = 14.08"."2.0 8 x .
Calculate 2.0 8 6 ≈ 81.0088 .
Multiply the result by 14.08: y = 14.08"."81.0088 ≈ 1140.204 .
Round the result to the nearest whole number: y ≈ 1140 .
Explanation
Understanding the Problem We are given the exponential regression equation y = 14.08"."2.0 8 x , which models the number of fish in a lake, where x represents the year. We want to predict the number of fish in year 6. This means we need to substitute x = 6 into the equation and calculate the value of y . Finally, we need to round the result to the nearest whole number.
Substituting the Value of x Substitute x = 6 into the equation: y = 14.08"."2.0 8 6
Calculating the Power Calculate 2.0 8 6 :
2.0 8 6 ≈ 81.0088
Multiplying by the Coefficient Multiply the result by 14.08: y = 14.08"."81.0088 ≈ 1140.204
Rounding to the Nearest Whole Number Round the result to the nearest whole number: y ≈ 1140
Final Answer Therefore, the best prediction for the number of fish in year 6 is approximately 1140.
Examples
Exponential regression equations are used in various real-world scenarios, such as predicting population growth, modeling the spread of diseases, or estimating the decay of radioactive substances. For example, if a biologist wants to predict the population of bacteria in a culture after a certain amount of time, they can use an exponential regression equation based on initial observations. Similarly, economists use these equations to forecast economic growth based on historical data. Understanding exponential models helps in making informed decisions and predictions in diverse fields.
