Question

Math

Posted 3 months ago

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After the last ice age began, the number of animal species in Australia changed rapidly.
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:
$S_{\text {year }}(t)=25,000,000 \cdot(0.78)^{t}$
Complete the following sentence about the rate of change in the number of species in decades.
Round your answer to two decimal places.
Every decade, the number of species decays by a factor of $\square$
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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

To find the rate of change in the number of species per decade, we need to calculate the value of the function $S_{\text{year}}(t)$ at $t=10$ years, since one decade is equivalent to 10 years

step 2

The function given is $S_{\text{year}}(t)=25,000,000 \cdot(0.78)^{t}$. We substitute $t=10$ into the function to find $S_{\text{year}}(10)$

step 3

Using the asksia-ll calculation list, we have already simplified $(25,000,000 \cdot 0.78^{10}) / 25,000,000$ to approximately $0.08336$. This represents the factor by which the number of species decays after 10 years

step 4

To express this decay factor for a decade, we round $0.08336$ to two decimal places, obtaining $0.08$

[question 1] Answer

Every decade, the number of species decays by a factor of $0.08$.

Key Concept

Exponential Decay

Explanation

The number of species is modeled by an exponential decay function, where the base of the exponent represents the decay factor per unit of time. After 10 years, or one decade, this factor is calculated and rounded to two decimal places to represent the rate of change in the number of species.

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