Explanation Given: The rate of increase of population is proportional to the number of people present at time t , the initial population has doubled in 5 yeras and the initial population is P 0 . The population after 3 years is 10000 . The time required for the population to quadruple is 10 years . Procedure used: (1) Form a differential equation. (2) Integrate the differential equation. (3) Apply initial conditions. Calculation: Form the differential equation from the given information. d P d t = k P d P P = k d t (1) Here, the population is P , the constant is k and the time is t . Integrate equation (1). ∫ d P P = ∫ k d t ln P = k t + c P = e k t e c (2) Substitute P 0 for P and 0 for t in equation (2). P 0 = e k ⋅ 0 e c P 0 = e c Substitute P 0 for e c in equation (2). P = P 0 e k t (3) Substitute 2 P 0 for P and 5 for t in equation (3). 2 P 0 = P 0 e k ⋅ 5 2 = e k ⋅ 5 5 k = ln 2 k = 1 5 ln 2 Substitute 1 5 ln 2 for k in equation (3). P = P 0 e 1 5 t ln 2 P = P 0 e ln 2 t 5 P = P 0 2 t 5 (4) Substitute 10000 for P and 3 for t in equation (4)

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 2.7, Problem 2E

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Determine the population after 10 years and the rate of increase of population.

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Chapter 2.7, Problem 1E

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For the Big-M tableau (of a maximization LP and row0 at bottom and M=1000), Z Ꮖ 1 x2 x3 81 82 83 e4 a4 RHS 0 7 0 0 1 0 4 3 -3 20 0 -4.5 0 0 0 1 -8 -2.5 2.5 6 0 7 0 1 0 0 8 3 -3 4 0 -1 50 1 0 0 0-2 -1 1 4 0000 0 30 970 200 If the original value of c₁ is increased by 60, what is the updated value of c₁ (meaning keeping the same set for BV. -10? Having made that change, what is the new optimal value for ž?

Here is the optimal tableau for a standard Max problem. zx1 x2 x3 24 81 82 83 rhs 1 0 5 3 0 6 0 1 .3 7.5 0 - .1 .2 0 0 28 360 0 -8 522 0 2700 0 6 12 1 60 0 0 -1/15-3 1 1/15 -1/10 0 2 Using that the dual solution y = CBy B-1 and finding B = (B-¹)-¹ we find the original CBV and rhs b. The allowable increase for b₂ is If b₂ is increased by 3 then, using Dual Theorem, the new value for * is If c₂ is increased by 10, then the new value for optimal > is i.e. if no change to BV, then just a change to profit on selling product 2. The original coefficients c₁ = =☐ a and c4 = 5 If c4 is changed to 512, then (first adjusting other columns of row0 by adding Delta times row belonging to x4 or using B-matrix method to update row0) the new optimal value, after doing more simplex algorithm, for > is

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