Question

this table shows the population of foxes in the ogden wildlife refuge every 5 years after 2010. the following function represents the population of rabbits in the ogden wildlife refuge x years after 2010. f(x) = 4,000 · 0.9^{x} complete the sentence. in 2010, there were rabbits than foxes. over time, the number of foxes while the number of rabbits.

Explanation:

Step1: Find the number of rabbits and foxes in 2010

For foxes, when \(x = 0\) (2010), from the table, the population of foxes \(y_f=200\).
For rabbits, use the function \(f(x)=4000\cdot0.9^x\). When \(x = 0\), \(f(0)=4000\cdot0.9^0\). Since \(a^0 = 1\) (\(a
eq0\)), \(f(0)=4000\times1 = 4000\).
The difference between the number of rabbits and foxes in 2010 is \(4000 - 200=3800\), so there were more rabbits than foxes.

Step2: Analyze the trend of fox - population

From the table:
When \(x = 0\), \(y_f=200\); when \(x = 5\), \(y_f = 400\); when \(x=10\), \(y_f=800\); when \(x = 15\), \(y_f=1600\); when \(x=20\), \(y_f=3200\).
We can see that \(y_f\) follows a pattern \(y_f=200\times2^{\frac{x}{5}}\) (exponential growth).

Step3: Analyze the trend of rabbit - population

For the function \(f(x)=4000\cdot0.9^x\), since the base of the exponential function \(b = 0.9<1\), the function \(y = a\cdot b^x\) (\(a>0\)) represents exponential decay.

Answer:

In 2010, there were more rabbits than foxes. Over time, the number of foxes increases while the number of rabbits decreases.