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_("year ")(t), is modeled by the following function: S_("year ")(t)=25,000,000*(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

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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_("year ")(t), is modeled by the following function:  S_("year ")(t)=25,000,000*(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

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Identify Function and Value: Identify the given function and the value to find. The function given is S_(＂year ＂)(t) = 25,000,000 * (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. 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. S_(＂year ＂)(10) = 25,000,000 * (0.78)^(10)Perform Calculation for t = 10: Perform the calculation for t = 10. S_(＂year ＂)(10) = 25,000,000 * (0.78)^(10) First, calculate (0.78)^(10). (0.78)^(10) ≈ 0.1073741824Multiply to Find Species: Multiply the result by 25,000,000 to find the number of species after one decade. S_(＂year ＂)(10) ≈ 25,000,000 * 0.1073741824 S_(＂year ＂)(10) ≈ 2,684,354.56Determine 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_(＂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 ≈ 2,684,354.56 / 25,000,000Perform Division: Perform the division to find the decay factor. Decay factor ≈ 2,684,354.56 / 25,000,000 Decay factor ≈ 0.1073741824Round Decay Factor: Round the decay factor to two decimal places. Decay factor ≈ 0.11 (rounded to two decimal places)

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