Explanation Given information: A paper drinking cup filled with water has the shape of a cone with height h and semi vertical angle θ . A ball is placed carefully in the cup, thereby displacing some of the water and making it overflow. Calculation: Sketch the A paper drinking cup filled with water has the shape of a cone with height h and semi vertical angle θ as shown in Figure 1. Refer to Figure 1. The cap of a sphere with radius r and height ( r + d ) . sin θ = r h − d h − d = r sin θ d = h − r csc θ Find the volume of the water displaced as shown below. V = 1 3 π h 2 ( 3 r − h ) Substitute ( r = d ) for h in the above Equation. V = 1 3 π ( r + d ) 2 ( 3 r − ( r + d ) ) = 1 3 π ( r + d ) 2 ( 2 r − d ) Substitute h − r csc θ for d in the above Equation. V = 1 3 π ( r + h − r csc θ ) 2 ( 2 r − ( h − r csc θ ) ) = 1 3 π ( r ( 1 − csc θ ) + h ) 2 ( r ( 2 + csc θ ) − h ) Differentiate both the sides of the Equation with respect to r . d V d r = π 3 [ ( r ( 1 − csc θ ) + h ) 2 ( 2 + csc θ ) + ( r ( 2 + csc θ ) − h ) [ 2 ( r ( 1 − csc θ

Bundle: Single Variable Calculus: Early Transcendentals, Loose-leaf Version, 8th + Webassign Printed Access Card For Calculus, Multi-term Courses, Life Of Edition

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ISBN: 9780357008034

Author: Stewart

Publisher: CENGAGE L

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Chapter 6, Problem 10P

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To calculate: The radius of the ball that causes the greatest volume of water to spill out of the cup.

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Chapter 6, Problem 9P

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Chapter 6 Solutions

Bundle: Single Variable Calculus: Early Transcendentals, Loose-leaf Version, 8th + Webassign Printed Access Card For Calculus, Multi-term Courses, Life Of Edition

Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Prob. 7E Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

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The radius of the ball that causes the greatest volume of water to spill out of the cup.

Solution Summary: The author calculates the radius of the ball that causes the greatest volume of water to spill out of a paper drinking cup.

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To calculate: The radius of the ball that causes the greatest volume of water to spill out of the cup.

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Chapter 6 Solutions