(a) Explanation Proof: Given information: The given function is, R ( t ) = 0.00731 t 4 − 0.174 t 3 + 1.528 t 2 + 0.48 t + 19.3 (1) Where, R ( t ) is the dependency ratio of the world population over the next century forecast and t is time in decade. The given interval of the function R ( t ) is ( 0 , 6 ) . Formula used: The formula to calculate the roots of a quadratic equation a t 2 + b t + c = 0 is, t = − b ± b 2 − 4 a c 2 a (2) Calculation: Differentiate equation (1) with respect to t . d d t [ R ( t ) ] = d d t ( 0.00731 t 4 − 0.174 t 3 + 1.528 t 2 + 0.48 t + 19.3 ) R ′ ( t ) = d d t ( 0.00731 t 4 ) − d d t ( 0.174 t 3 ) + d d t ( 1.528 t 2 ) + d d t ( 0.48 t ) + d d t ( 19.3 ) Further solve the above equation. R ′ ( t ) = 0.00731 × 4 t 3 − 0.174 × 3 t 2 + 1.528 × 2 t + 0.48 + 0 ( ∵ d x n d x = n x n − 1 and d ( c ) d x = 0 ) = 0.02924 t 3 − 0.522 t 2 + 3.056 t + 0.48 Differentiate above equation with respect to t to get R ″ ( t ) (b) To determine To find: The dependency ratio of the world population by the given function when it will be increasing at the fastest pace.

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 4.2, Problem 103E

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To show: The dependency ratio of the world population by the given function will be increasing at the fastest pace in the year 2052.

(b)

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To find: The dependency ratio of the world population by the given function when it will be increasing at the fastest pace.

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Chapter 4.2, Problem 102E

Chapter 4.2, Problem 104E

Chapter 4 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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In this application, we set up a mathematical model for determining the total costs in setting up a training program, such as a hospital might use. Then we use calculus to find the time interval between training programs that produces the minimum total cost. The model assumes that the demand for trainees is constant and that the fixed cost of training a batch of trainees is known. Also, it is assumed that people who are trained, but for whom no job is readily available, will be paid a fixed amount per month while waiting for a job to open up. The model uses the following variables. D = demand for trainees per month N = number of trainees per batch C1 = Fixed cost of training a batch of trainees C2 = marginal cost of training per trainee per month C3 = salary paid monthly to a trainee who has not yet been given a job after training m = time interval in months between successive batches of trainees t = length of training program in months Z(m) = total monthly cost of program The total cost of training a batch of trainees is given by C1+NtC2. However, N=mD, so that the total cost per batch is C1+mDtC2. After training, personnel are given jobs at the rate of D per month. Thus, ND of the trainees will not get a job the first month, N2D will not get a job the second month, and so on. The ND trainees who do not get a job the first month produce total costs of (ND)C3, and so on. Since N=mD, the costs during the first month can be written as. (ND)C3=(mDD)C3=(m1)DC3 While the costs during the second month are (m2)DC3, and so on. The total cost for keeping the trainees without a job is thus (m1)DC3+(m2)DC3+(m3)DC3++2DC3+DC3 Which can be factored to give DC3[(m1)+(m2)+(m3)++2+1] The expression in bracket is the sum of the terms of an arithmetic sequence, discussed in most algebra texts. Using formulas for arithmetic sequences, the expression in brackets can be shown to equal m(m1)2, so that we have DC3[m(m1)2](1) As the total cost for keeping jobless trainees. The total cost per batch is the sum of the training cost per batch. C1+mDtC2, and the cost of keeping trainees without a proper job, given by equation 1, since we assume that a batch of trainees is trained every m months, the total cost per month, Z(m), is given by Z(m)=C1+mDtC2m+DC3[m(m1)2]m=C1m+DtC2+DC3(m12) Source: P.I., Goyal and S.K. Goyal. Find Z(m)

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The dependency ratio of the world population by the given function will be increasing at the fastest pace in the year 2052.

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