Question

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the function $f(x) = 1200(1.055)^x$ models the balance of an investment $x$ years after it is made.
how does the average rate of change between years 21 and 25 compare to the average rate of change between years 1 and 5?

• the average rate of change between years 21 and 25 is about 2 times the rate between years 1 and 5.
• the average rate of change between years 21 and 25 is about $\frac{1}{5}$ the rate between years 1 and 5.
• the average rate of change between years 21 and 25 is about $\frac{1}{4}$ the rate between years 1 and 5.
• the average rate of change between years 21 and 25 is about 3 times 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 for [1, 5]

For \( f(x) = 1200(1.055)^x \), over \([1, 5]\):
\( f(1) = 1200(1.055)^1 = 1200\times1.055 = 1266 \)
\( f(5) = 1200(1.055)^5 \approx 1200\times1.30696 \approx 1568.35 \)
Average rate \( r_1=\frac{f(5)-f(1)}{5 - 1}=\frac{1568.35 - 1266}{4}=\frac{302.35}{4}\approx75.59 \)

Step3: Calculate Average Rate for [21, 25]

Over \([21, 25]\):
\( f(21)=1200(1.055)^{21}\approx1200\times3.0502\approx3660.24 \)
\( f(25)=1200(1.055)^{25}\approx1200\times3.7975\approx4557 \)
Average rate \( r_2=\frac{f(25)-f(21)}{25 - 21}=\frac{4557 - 3660.24}{4}=\frac{896.76}{4}\approx224.19 \)

Step4: Find Ratio of \( r_2 \) to \( r_1 \)

\(\frac{r_2}{r_1}\approx\frac{224.19}{75.59}\approx3\) (or use exponent property: \(\frac{(1.055)^{25}-(1.055)^{21}}{(1.055)^5-(1.055)^1}=\frac{(1.055)^{21}((1.055)^4 - 1)}{(1.055)^1((1.055)^4 - 1)}=(1.055)^{20}\approx(1.055^5)^4\approx(1.30696)^4\approx3\))

Answer:

The average rate of change between years 21 and 25 is about 3 times the rate between years 1 and 5.