A melittologist measures the population P(t) of a bee hive after t months and records the results below. 2 4 6. 8. 10 P(t) 5000 11600 22500 34500 42900 47100 (a) For t = 2, 4, 6, 8, find the slope of the data immediately before and after each point to estimate P'(2), P(4), P'(6), and P'(8) (for example, P'(2) - 2-P0)). What are the units of P'(t)? What do these values tell you about the shape of the graph of P(t) vs. t? 4-0 (b) Populations with a maximum carrying capacity are modeled using the logistic equation: P'(t) = k P(t) (C – P(t)) where k is the growth constant and C is the carrying capacity. Using the population and population growth from part (a), calculate 8 for t = 2,4, 6, 8 (rounded to three decimals). Let X = Y vs. X plot of the four points when t = 2, 4, 6, 8. What pattern do you notice about the four points? Find the equation of the line that passes through the first point and last point in terms of Y English sentence, explain why the population is at the carrying capacity C when Y the carrying capacity C by setting Y equal to zero in the equation Y = mX + b and solving for X. Then plug in C, P(2), and P'(2) into the logistic equation and solve for the growth constant k. This process is known as parametrizing the model. P(t) and Y P'(t) P(t) and make a P(t) mX + b. In a complete PO equals zero. Find
A melittologist measures the population P(t) of a bee hive after t months and records the results below. 2 4 6. 8. 10 P(t) 5000 11600 22500 34500 42900 47100 (a) For t = 2, 4, 6, 8, find the slope of the data immediately before and after each point to estimate P'(2), P(4), P'(6), and P'(8) (for example, P'(2) - 2-P0)). What are the units of P'(t)? What do these values tell you about the shape of the graph of P(t) vs. t? 4-0 (b) Populations with a maximum carrying capacity are modeled using the logistic equation: P'(t) = k P(t) (C – P(t)) where k is the growth constant and C is the carrying capacity. Using the population and population growth from part (a), calculate 8 for t = 2,4, 6, 8 (rounded to three decimals). Let X = Y vs. X plot of the four points when t = 2, 4, 6, 8. What pattern do you notice about the four points? Find the equation of the line that passes through the first point and last point in terms of Y English sentence, explain why the population is at the carrying capacity C when Y the carrying capacity C by setting Y equal to zero in the equation Y = mX + b and solving for X. Then plug in C, P(2), and P'(2) into the logistic equation and solve for the growth constant k. This process is known as parametrizing the model. P(t) and Y P'(t) P(t) and make a P(t) mX + b. In a complete PO equals zero. Find
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Transcribed Image Text:A melittologist measures the population P(t) of a bee hive after t months and records the results below.
2
4
6.
8.
10
P(t)
5000
11600
22500
34500
42900
47100
(a) For t = 2, 4, 6, 8, find the slope of the data immediately before and after each point to estimate P'(2), P(4), P'(6),
and P'(8) (for example, P'(2) - 2-P0)). What are the units of P'(t)? What do these values tell you about
the shape of the graph of P(t) vs. t?
4-0
(b) Populations with a maximum carrying capacity are modeled using the logistic equation:
P'(t) = k P(t) (C – P(t))
where k is the growth constant and C is the carrying capacity. Using the population and population growth from
part (a), calculate 8 for t = 2,4, 6, 8 (rounded to three decimals). Let X =
Y vs. X plot of the four points when t = 2, 4, 6, 8. What pattern do you notice about the four points? Find the
equation of the line that passes through the first point and last point in terms of Y
English sentence, explain why the population is at the carrying capacity C when Y
the carrying capacity C by setting Y equal to zero in the equation Y = mX + b and solving for X. Then plug
in C, P(2), and P'(2) into the logistic equation and solve for the growth constant k. This process is known as
parametrizing the model.
P(t) and Y
P'(t)
P(t)
and make a
P(t)
mX + b. In a complete
PO equals zero. Find
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