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Posted 3 months ago

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

The relationship between the elapsed time, tt, in years, since the ice age began, and the total number of animal species, Syear (t)S_{\text {year }}(t), is modeled by the following function:
Syear (t)=25,000,000(0.78)t
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 Syear(t)S_{\text{year}}(t) at t=10t=10 years, since one decade is equivalent to 10 years
step 2
The function given is Syear(t)=25,000,000(0.78)tS_{\text{year}}(t)=25,000,000 \cdot(0.78)^{t}. We substitute t=10t=10 into the function to find Syear(10)S_{\text{year}}(10)
step 3
Using the asksia-ll calculation list, we have already simplified (25,000,0000.7810)/25,000,000(25,000,000 \cdot 0.78^{10}) / 25,000,000 to approximately 0.083360.08336. 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 0.083360.08336 to two decimal places, obtaining 0.080.08
[question 1] Answer
Every decade, the number of species decays by a factor of 0.080.08.
Key Concept
Exponential Decay
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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