(a) Explanation Given: The fertility rate in EU remains at 1.8 until 2020. The population of the 15 member countries of the EU is approximated over the next century by P ( t ) = 325 + 7.475 ( t + 10 ) e − t + 10 20 , where t is the number of years since 2000. Calculation: The given function is P ( t ) = 325 + 7.475 ( t + 10 ) e − t + 10 20 . To find the year with the largest population, obtain the derivative of the function P ( t ) = 325 + 7.475 ( t + 10 ) e − t + 10 20 with respect to t as follows. P ′ ( t ) = ( 325 + 7.475 ( t + 10 ) e − t + 10 20 ) ′ = 7.475 [ ( t + 10 ) ( e − t + 10 20 ) ′ + e − t + 10 20 ( t + 10 ) ′ ] = 7.475 [ ( t + 10 ) ( e − t + 10 20 ) ⋅ ( − 1 20 ) + e − t + 10 20 ] = 7.475 e − t + 10 20 ( − t 20 − 1 2 + 1 ) Simplify further as follows. P ′ ( t ) = 7.475 e − t + 10 20 ( − t 20 − 1 2 + 1 ) = 7.475 e − t + 10 20 ( − t 20 + 1 2 ) Set P ′ ( c ) = 0 and obtain the value of c . 7.475 e − c + 10 20 ( − c 20 + 1 2 ) = 0 e − c + 10 20 = 0 or − c 20 + 1 2 = 0 − c 20 + 1 2 = 0 c = 10 Consider two numbers on either side of the value c = 10 and obtain the sign of the derivative at these points (b) To determine To find: The year in which the population declined. (c) To determine To find: The value to which the population approaches as time goes on.

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 13, Problem 74RE

(a)

To determine

To find: The year in which the population can be the largest and the population in that year.

(b)

To determine

To find: The year in which the population declined.

(c)

To determine

To find: The value to which the population approaches as time goes on.

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Chapter 13, Problem 73RE

Chapter 13, Problem 75RE

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Finite Mathematics and Calculus with Applications (10th Edition)

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The year in which the population can be the largest and the population in that year.

Solution Summary: The author explains that the fertility rate in the EU remains at 1.8 until 2020. To find the year with the largest population, obtain the derivative of the function with respect to t.

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To find: The year in which the population can be the largest and the population in that year.

To find: The year in which the population declined.

To find: The value to which the population approaches as time goes on.

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Chapter 13 Solutions