The population of butterflies in a certain nature center is known to grow according to the logistic model. Let B(t) be the number of butterflies in the nature center where t is measured in years after 2010 (so t = 0 represents the year 2010). Suppose we know B(0) = 200, B(1) = 300, and as t → ∞, B(t) → 1000. (a) Determine the exact values of A, M, and k in the logistic model B(t) = A. 1 + Me−kt (b) Use graphing technology to graph y = B(t). (c) Find the exact value of B(2) and then use a calculator to approximate it to 2 decimal places. Include units and write a sentence to explain what the value tells us about the butterfly population. (d) Find the exact value of t for which B(t) = 800 and then use a calculator to approximate it to 2 decimal places. Include units and write a sentence to explain what the value tells us about the butterfly population.
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The population of butterflies in a certain nature center is known to grow according to the logistic model. Let B(t) be the number of butterflies in the nature center where t is measured in years after 2010 (so t = 0 represents the year 2010). Suppose we know B(0) = 200, B(1) = 300, and as t → ∞, B(t) → 1000.
(a) Determine the exact values of A, M, and k in the logistic model B(t) = A.
1 + Me−kt (b) Use graphing technology to graph y = B(t).
(c) Find the exact value of B(2) and then use a calculator to approximate it to 2 decimal places. Include units and write a sentence to explain what the value tells us about the butterfly population.
(d) Find the exact value of t for which B(t) = 800 and then use a calculator to approximate it to 2 decimal places. Include units and write a sentence to explain what the value tells us about the butterfly population.
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