Question

calculator
the function $f(x) = 125(0.9)^x$ models the population of a species of fly in millions after $x$ years.
how does the average rate of change between years 11 and 15 compare to the average rate of change between years 1 and 5?
\bigcirc\\ the average rate of change between years 11 and 15 is about 2 times the rate between years 1 and 5.
\bigcirc\\ the average rate of change between years 11 and 15 is about 3 times the rate between years 1 and 5.
\bigcirc\\ the average rate of change between years 11 and 15 is about $\frac{1}{2}$ the rate between years 1 and 5.
\bigcirc\\ the average rate of change between years 11 and 15 is about $\frac{1}{3}$ the rate between years 1 and 5.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\).

Step2: Calculate average rate of change for [1, 5]

First, find \( f(1) \) and \( f(5) \) for \( f(x) = 125(0.9)^x \).

• \( f(1) = 125(0.9)^1 = 125 \times 0.9 = 112.5 \)
• \( f(5) = 125(0.9)^5 \). Calculate \( (0.9)^5 \approx 0.59049 \), so \( f(5) \approx 125 \times 0.59049 \approx 73.81125 \)

The average rate of change for \([1, 5]\) is \(\frac{f(5) - f(1)}{5 - 1} = \frac{73.81125 - 112.5}{4} = \frac{-38.68875}{4} \approx -9.6721875\)

Step3: Calculate average rate of change for [11, 15]

Find \( f(11) \) and \( f(15) \).

• \( f(11) = 125(0.9)^{11} \). Calculate \( (0.9)^{11} \approx 0.313810596 \), so \( f(11) \approx 125 \times 0.313810596 \approx 39.2263245 \)
• \( f(15) = 125(0.9)^{15} \). Calculate \( (0.9)^{15} \approx 0.205891132 \), so \( f(15) \approx 125 \times 0.205891132 \approx 25.7363915 \)

The average rate of change for \([11, 15]\) is \(\frac{f(15) - f(11)}{15 - 11} = \frac{25.7363915 - 39.2263245}{4} = \frac{-13.489933}{4} \approx -3.37248325\)

Step4: Compare the two rates

Divide the average rate of change for \([11, 15]\) by that for \([1, 5]\): \(\frac{-3.37248325}{-9.6721875} \approx \frac{1}{3}\) (since both are negative, the negative signs cancel, and the ratio is approximately \(\frac{1}{3}\)). So the average rate of change between years 11 and 15 is about \(\frac{1}{3}\) the rate between years 1 and 5.

Answer:

The average rate of change between years 11 and 15 is about \(\frac{1}{3}\) the rate between years 1 and 5. (The option stating this, e.g., "The average rate of change between years 11 and 15 is about \(\frac{1}{3}\) the rate between years 1 and 5.")