Question

the fish population in a lake can be modeled by the function ( p(x) = 3,640(1.15)^x ), where ( x ) represents the number of years since 2000. which statement best interprets one value in the function?
a) the fish population increases at a rate of 36.40% per year.
b) the initial fish population is 4,186.
c) the initial fish population is 115.
d) the fish population increases at a rate of 15% per year.

Explanation:

The function for the fish population is \( P(x)=3640(1.15)^x \), where \( x \) is the number of years since 2000. In an exponential growth model of the form \( P(x)=a(1 + r)^x \), \( a \) is the initial population and \( r \) is the growth rate (expressed as a decimal). Here, \( a = 3640 \) (initial population) and \( 1 + r=1.15 \), so \( r = 0.15 \) or 15% growth per year.

• Option a: The growth rate from \( 1.15 \) is 15%, not 36.40%, so a is incorrect.
• Option b: The initial population \( a \) is 3640, not 4186, so b is incorrect.
• Option c: The initial population \( a \) is 3640, not 115, so c is incorrect.
• Option d: Since \( 1 + r = 1.15 \), \( r=0.15 = 15\% \), so the fish population increases at 15% per year, making d correct.

Answer:

d. The fish population increases at a rate of 15% per year.