After the last ice age began, the number of animal species in Australia changed rapidly. $\newline$ The relationship between the elapsed time, $t$ , in years, since the ice age began, and the total number of animal species, $S_{\text {year }}(t)$ , is modeled by the following function: $\newline$ $S_{\text {year }}(t)=25,000,000 \cdot(0.78)^{t}$ $\newline$ Complete the following sentence about the rate of change in the number of species in decades. Round your answer to two decimal places. $\newline$ Every decade, the number of species decays by a factor of

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Q. After the last ice age began, the number of animal species in Australia changed rapidly.

\newline

The relationship between the elapsed time,

t

, in years, since the ice age began, and the total number of animal species,

S_{\text {year }}(t)

, is modeled by the following function:

\newline

S_{\text {year }}(t)=25,000,000 \cdot(0.78)^{t}

\newline

Complete the following sentence about the rate of change in the number of species in decades. Round your answer to two decimal places.

\newline

Every decade, the number of species decays by a factor of

Identify Function and Value: Identify the given function and the value to find. $\newline$ The function given is $S_{\text{year}}(t) = 25,000,000 \times (0.78)^t$ , where $t$ is the time in years since the ice age began. We need to find the decay factor for every decade ( $10$ years).
Calculate Decay Factor: Calculate the decay factor for one decade. $\newline$ To find the decay factor for one decade, we need to substitute $t = 10$ into the function because one decade is equivalent to $10$ years. $\newline$ $S_{\text{year}}(10) = 25,000,000 \times (0.78)^{10}$
Perform Calculation for $t = 10$ : Perform the calculation for $t = 10$ . $S_{\text{year}}(10) = 25,000,000 \times (0.78)^{10}$ First, calculate $(0.78)^{10}$ . $(0.78)^{10} \approx 0.1073741824$
Multiply to Find Species: Multiply the result by $25,000,000$ to find the number of species after one decade. $\newline$ $S_{\text{year }}(10) \approx 25,000,000 \times 0.1073741824$ $\newline$ $S_{\text{year }}(10) \approx 2,684,354.56$
Determine Decay Factor: Determine the decay factor by comparing the number of species at $t = 0$ and $t = 10$ . Initially, at $t = 0$ , the number of species is $S_{\text{year}}(0) = 25,000,000$ . After one decade, the number of species is approximately $2,684,354.56$ . To find the decay factor, we divide the number of species after one decade by the initial number of species. Decay factor $\approx 2,684,354.56 / 25,000,000$
Perform Division: Perform the division to find the decay factor. $\newline$ Decay factor $\approx \frac{2,684,354.56}{25,000,000}$ $\newline$ Decay factor $\approx 0.1073741824$
Round Decay Factor: Round the decay factor to two decimal places. $\newline$ Decay factor $\approx 0.11$ (rounded to two decimal places)