The number of fish, f, in Skipper's Pond at the beginning of each year can be modeled by the equation [tex]$f ( x )=3\left(2^{ x }\right)$[/tex], where x represents the number of years after the beginning of the year 2000. For example, [tex]$x =0$[/tex] represents the beginning of the year 2000, [tex]$x =1$[/tex] represents the beginning of the year 2001, and so forth. According to the model, how many fish were in Skipper's Pond at the beginning of the year 2006?
A. 96
B. 192
C. 384
D. 1,458
E. 46,656
Answer (1)
Determine the value of x for the year 2006: x = 2006 − 2000 = 6 .
Substitute x into the equation: f ( 6 ) = 3 ( 2 6 ) .
Calculate 2 6 = 64 .
Calculate the number of fish: f ( 6 ) = 3 × 64 = 192 .
Explanation
Understanding the Problem We are given the equation f ( x ) = 3 ( 2 x ) that models the number of fish in Skipper's Pond at the beginning of each year, where x is the number of years after the beginning of the year 2000. We want to find the number of fish at the beginning of the year 2006.
Finding the Value of x First, we need to determine the value of x that corresponds to the year 2006. Since x represents the number of years after the beginning of 2000, we can calculate x as follows: x = 2006 − 2000 = 6
Calculating the Number of Fish Now, we substitute x = 6 into the equation f ( x ) = 3 ( 2 x ) to find the number of fish at the beginning of 2006: f ( 6 ) = 3 ( 2 6 ) We know that 2 6 = 64 , so f ( 6 ) = 3 ( 64 ) = 192
Final Answer Therefore, according to the model, there were 192 fish in Skipper's Pond at the beginning of the year 2006.
Examples
Understanding exponential growth, as modeled by the fish population in Skipper's Pond, can be applied to various real-world scenarios. For instance, it can help in predicting the spread of a virus, the growth of an investment, or the decay of radioactive material. By using the formula f ( x ) = 3 ( 2 x ) , we can estimate future values based on an initial amount and a constant growth rate. This type of modeling is crucial in fields like epidemiology, finance, and environmental science, allowing for informed decision-making and strategic planning. For example, if an investment doubles every year, starting with an initial investment of $3, after 6 years, the investment would be $3 \times 2^6 = $192.
