Explanation Given: D ( r ) = 25 , 000 e − 0.05 r 2 people/mi 2 . Calculation: Consider the following function that gives the population density r miles from the center of a certain city, D ( r ) = 25 , 000 e − 0.05 r 2 people per square mile ...... (1) For 1 mile from the city center, r = 1 For 2 miles from the city center, r = 2 The area of the population that live r miles from the city center will have the shape of a circle with radius r Divide the interval 1 ≤ r ≤ 2 into n equal subintervals of width Δ r miles. Assume that r j represents the beginning of j t h sub-interval, then n sub-intervals determine n concentric rings. Area of the j t h ring will be equal to the product of circumference of the ring and its width. That is, Area of j th ring = circumference of ring × width =2 π r j × Δ r The population in the j t h ring will be equal to the product of area of the ring and the population density function D ( r j ) . That is, Population in j th ring = area of ring × population density function =2 π r j Δ r × D ( r j ) The population between 1 and 2 miles from the city center will be the sum of n such terms, one for each of the n concentric rings. That is, Population between I and 2 miles from the city center = lim n → ∞ ∑ j = 1 n 2 π r j Δ r j D ( r j ) = ∫ 1 2 2 π r D ( r ) d r Substitute, the values in the above expression D ( r ) = 25 , 000 e − 0

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6.6, Problem 49PS

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To find: the number of people lives between 1 and 2 miles from the city center

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Chapter 6.6, Problem 48PS

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the number of people lives between 1 and 2 miles from the city center

Solution Summary: The author calculates the population density r miles from the city center by dividing the interval into n equal subintervals of width.

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