A bee hive has a sustainable population of 50 000 bees. The population P(t) is modeled by the differential equation P'(t) = 10-5 P(t) (50 000 – P(t)), where t is measured in mont hs. The amount of sugar each bee requires each month is 0.25 grams per bee per month. (a) The population of the hive is measured once every two months and recorded below: t (months) 4 6. 8. 10 P(t) (# of bees) 5000 11600 22500 34500 42900 47100 Sugar Rate (g/month) First find the rate of sugar consumption by the entire hive for each measurement. Then use the table to find the left Riemann sum with 5 rectangles, the right Riemann sum with 5 rectangles, and the average of the two to estimate the accumulated sugar consumption over ten months. Express your final answer in kilograms. Is this a reasonable amount of sugar under these circumstances? 50 000 (b) Show that P(t) satisfies the differential equation by first evaluating the left-hand side of the %3D 1+ 9e-.5t differential equation (i.e., finding the derivative P'(t) of P(t) = 0000), then evaluating the right-hand side of the differential equation (i.e., plugging in P(t) = 00 into 10-5 P(t) (50 000 – P(t)) and simplifying into a single fraction), and finally showing that the two results are the same. Then answer the following questions, including units for all answers. • What is the initial population? • Find the limiting population lim P(t). • Find the instantaneous rate of change of population at 1 month. • Find the inflection point of P(t) by finding what the population P is population growth P' maximized. What is the population growth P' at this population? (Hint: Do NOT find the second derivative of P(t). Instead, let f(P) = 10-5 P(50 000 P). Then find the marimum of this function f(P) with respect to P.)
A bee hive has a sustainable population of 50 000 bees. The population P(t) is modeled by the differential equation P'(t) = 10-5 P(t) (50 000 – P(t)), where t is measured in mont hs. The amount of sugar each bee requires each month is 0.25 grams per bee per month. (a) The population of the hive is measured once every two months and recorded below: t (months) 4 6. 8. 10 P(t) (# of bees) 5000 11600 22500 34500 42900 47100 Sugar Rate (g/month) First find the rate of sugar consumption by the entire hive for each measurement. Then use the table to find the left Riemann sum with 5 rectangles, the right Riemann sum with 5 rectangles, and the average of the two to estimate the accumulated sugar consumption over ten months. Express your final answer in kilograms. Is this a reasonable amount of sugar under these circumstances? 50 000 (b) Show that P(t) satisfies the differential equation by first evaluating the left-hand side of the %3D 1+ 9e-.5t differential equation (i.e., finding the derivative P'(t) of P(t) = 0000), then evaluating the right-hand side of the differential equation (i.e., plugging in P(t) = 00 into 10-5 P(t) (50 000 – P(t)) and simplifying into a single fraction), and finally showing that the two results are the same. Then answer the following questions, including units for all answers. • What is the initial population? • Find the limiting population lim P(t). • Find the instantaneous rate of change of population at 1 month. • Find the inflection point of P(t) by finding what the population P is population growth P' maximized. What is the population growth P' at this population? (Hint: Do NOT find the second derivative of P(t). Instead, let f(P) = 10-5 P(50 000 P). Then find the marimum of this function f(P) with respect to P.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A bee hive has a sustainable population of 50 000 bees. The population P(t) is modeled by the differential
equation
P'(t) = 10-5 P(t) (50 000 – P(t)),
where t is measured in mont hs. The amount of sugar each bee requires each month is 0.25 grams per bee per month.
(a) The population of the hive is measured once every two months and recorded below:
t (months)
0.
4
6.
8.
10
P(t) (# of bees)
5000
11600
22500
34500
42900
47100
Sugar Rate (g/month)
First find the rate of sugar consumption by the entire hive for each measurement. Then use the table to find
the left Riemann sum with 5 rectangles, the right Riemann sum with 5 rectangles, and the average of the two to
estimate the accumulated sugar consumption over ten months. Express your final answer in kilograms. Is this a
reasonable amount of sugar under these circumstances?
50 000
(b) Show that P(t) =
satisfies the differential equation by first evaluating the left-hand side of the
%3D
1+9e-.5t
50 000
differential equation (i.e., finding the derivative P'(t) of P(t) = 1 ), then evaluating the right-hand side
of the differential equation (i.e., plugging in P(t) = into 10-5 P(t) (50 000 – P(t)) and simplifying into
a single fraction), and finally showing that the two results are the same. Then answer the following questions,
including units for all answers.
%3D
50 000
• What is the initial population?
• Find the limiting population lim P(t).
• Find the instantaneous rate of change of population at 1 month.
• Find the inflection point of P(t) by finding what the population P is population growth P' maximized.
What is the population growth P' at this population? (Hint: Do NOT find the second derivative of P(t).
Instead, let f(P) = 10P(50000 - P). Then find the marimum of this function f(P) with respect to P.)
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