(a) Explanation Given information: The width of the satellite is 2 m. The depth of the satellite is 1 2 m . The axis of symmetry of the satellite is tilted 30 ° from the vertical. Calculation: Provide the equation of satellite in standard position as shown below. z = 1 2 x 2 + 1 2 y 2 Sketch the position of the satellite as shown in Figure 1. Refer to Figure 1. The unit vector normal to the plane of the water level is v = 0 i + a j + b k . The unit vector satisfies the following conditions. b = v ⋅ k = cos 30 ° = 3 2 Also, a = − 1 − b 2 = − 1 − ( 3 2 ) 2 = − 1 − 3 4 = − 1 4 = − 1 2 Provide the unit vector as shown below. v = − 1 2 j + 3 2 k ⇒ − 1 2 ( y − 1 ) + 3 2 ( z − 1 2 ) = 0 ( z − 1 2 ) = 1 3 y − 1 3 z = 1 3 y + ( 1 2 − 1 3 ) This is an equation of the plane of the water level. Provide the volume of the water as shown below. V = ∬ R ∫ 1 2 x 2 + 1 2 y 2 1 3 y + 1 2 − 1 3 d z d y d x Here, R is the interior of ellipse x 2 + y 2 − 2 3 y − 1 + 2 3 = 0 . If x = 0 , then y = α or y = β (b) To determine Calculate the smallest angle of the satellite so that it holds no water.

University Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780321999580

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14, Problem 26AAE

(a)

To determine

Provide a triple integral that provides the volume of water the satellite dish will hold.

(b)

To determine

Calculate the smallest angle of the satellite so that it holds no water.

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Chapter 14, Problem 25AAE

Chapter 14, Problem 27AAE

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Chapter 14 Solutions

University Calculus: Early Transcendentals (3rd Edition)

Ch. 14.1 - In Exercises 1-14. evaluate the iterated...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - Prob. 10E

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