Explanation Given: p ( r ) = 4 ( 1 + r 2 ) 1 3 (in thousands per square mile) Calculation: We have been given that, radius considered=10 mile and radial population density = p ( r ) = 4 ( 1 + r 2 ) 1 3 which varies from centre of city 0 to radius considered 10. To calculate the population in the given circle, thin rings inside the circle are considered, as shown below, Now, if we straightened out each ring we get rectangular strips, with length as the circumference of that ring 2πr (r is radius of the ring) and breadth as the difference in radius dr Therefore, area of ring = 2πr*dr square miles Therefore, Population of ring = area of ring* radial population density Therefore, total population, P in thousands = sum of (area of ring*radial population density) = integral of 2πrdr* radial population density p(r) (in thousands per square mile) P = ∫ 0 10 2 π r p ( r ) d r P = 2 π ∫ 0 10 r p ( r ) d r ∴ P = 2 π ∫ 0 10 r 4 ( 1 + r 2 ) 1 3 d r ∴ P = 2 π × 4 ∫ 0 10 r ( 1 + r 2 ) 1 3 d r ∴ P = 8 π ∫ 0 10 r ( 1 + r 2 ) 1 3 d r L e t , u = 1 + r 2 d u = 2 r d r ∴ r d r = d u 2 ∴ P = 8 π ∫ 0 10 u 1 3 d u 2 ∴ P = 8 π × 1 2 ∫ 0 10 u 1 3 d u ∴

4th Edition

ISBN: 9781319308858

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 6.2, Problem 29E

To determine

To calculate: the total population within a 10-mile radius of the city centre if the radial population density is $p (r) = 4 {(1 + r^{2})}^{\frac{1}{3}}$ (in thousands per square mile)

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Chapter 6.2, Problem 28E

Chapter 6.2, Problem 30E

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Chapter 6 Solutions

CALCULUS:EARLY TRANS.(LL)-W/ACCESS

Ch. 6.1 - Prob. 1PQ Ch. 6.1 - Prob. 2PQ Ch. 6.1 - Prob. 3PQ Ch. 6.1 - Prob. 4PQ Ch. 6.1 - Prob. 5PQ Ch. 6.1 - Prob. 6PQ Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E

Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - Prob. 13E Ch. 6.1 - Prob. 14E Ch. 6.1 - Prob. 15E Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - Prob. 42E Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - Prob. 47E Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - Prob. 51E Ch. 6.1 - Prob. 52E Ch. 6.1 - Prob. 53E Ch. 6.1 - Prob. 54E Ch. 6.1 - 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Knowledge Booster

the total population within a 10-mile radius of the city centre if the radial population density is p ( r ) = 4 ( 1 + r 2 ) 1 3 (in thousands per square mile)

To calculate: the total population within a 10-mile radius of the city centre if the radial population density is p(r)=4(1+r2)13(in thousands per square mile)

Want to see the full answer?

Chapter 6 Solutions

To calculate: the total population within a 10-mile radius of the city centre if the radial population density is $p (r) = 4 {(1 + r^{2})}^{\frac{1}{3}}$ (in thousands per square mile)