Question

the growth of a species of fish in a lake can be modeled by the function $f(x)=500(1.10)^x$, where $x$ is time in months since january 2019. what does the 500 represent? a the number of fish after the first month. b the monthly growth rate of the fish. c the number of fish at the end of the first year. d the starting number of fish in the lake.

Explanation:

The function \( f(x) = 500(1.10)^x \) is an exponential growth model. In the general form of an exponential growth function \( f(x)=a(b)^x \), \( a \) represents the initial amount (when \( x = 0 \)). Here, \( x \) is time in months since January 2019, so when \( x = 0 \) (at January 2019), \( f(0)=500(1.10)^0 = 500(1)=500 \). This means 500 is the starting number of fish. Option A is incorrect because after the first month (\( x = 1 \)), \( f(1)=500(1.10)^1 = 550 \), not 500. Option B is incorrect as the monthly growth rate is \( 1.10 - 1=0.10 \) (or 10%), not 500. Option C is incorrect because at the end of the first year (\( x = 12 \)), \( f(12)=500(1.10)^{12}\approx1569 \), not 500.

Answer:

D. The starting number of fish in the lake.