Explanation Given: The cone is x 2 + y 2 ≤ h ≤ 3 and the mass density is δ ( h ) = a e − b h where a = 1.225 × 10 9 and b = 0.13 Formula used: The total mass of a cone with a given mass density δ ( h ) is the solution of the integration ∫ h A δ ( h ) d h where d h is an elementary height and A is the elementary area of thecone. Calculation: Here the region is the given cone, So, if we cut the cone by a horizontal plane then we will get a circular disk x 2 + y 2 = h 2 whose radius will be h and area will be π h 2 . Then we integrate the elementary area A = π h 2 on the range of h multiplying by the density function δ ( h ) to get the total mass of the given cone. Now 0 ≤ h ≤ 3 (only positive side of h is given here), so, the integration will be ∫ h A δ ( h ) d h = ∫ 0 3 π h 2 a e − b h d h = a π ∫ 0 3 h 2 e − b h d h = a π [ h 2 e − b h − b ] 0 3 + 2 a π

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.5, Problem 9E

To determine

The total mass of a cone shaped region assuming a mass density $δ (h)$ depending on height $h$ .

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Chapter 16.5, Problem 8E

Chapter 16.5, Problem 10E

Chapter 16 Solutions

CALCULUS (CLOTH)

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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The total mass of a cone shaped region assuming a mass density δ ( h ) depending on height h .

Solution Summary: The author calculates the total mass of the cone shaped region assuming a mass density of (h) depending on height h.

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The total mass of a cone shaped region assuming a mass density $δ (h)$ depending on height $h$ .

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Chapter 16 Solutions

The total mass of a cone shaped region assuming a mass density δ ( h ) depending on height h .

Solution Summary: The author calculates the total mass of the cone shaped region assuming a mass density of (h) depending on height h.

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The total mass of a cone shaped region assuming a mass density δ(h) depending on height h.

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Chapter 16 Solutions

The total mass of a cone shaped region assuming a mass density $δ (h)$ depending on height $h$ .