Explanation Find the radius of the washer that is obtained by the intersection of the cylinder with the sphere. Then determine the volume of the cross-section. Given: V = π 6 h 3 . Since V can be expressed in terms of h alone, it follows that two beads of height 1 cm, one formed from a sphere the size of an orange and the other from a sphere the size of the earth would have the same volume. Calculation: The equation for the outer circle of the bead is given by x 2 + y 2 = R 2 and the inner cylinder intersects the sphere when y = ± R 2 − r 2 . Each cross-section of the bead will be a washer with outer radius given by R 2 − y 2 and inner radius r . The volume will be therefore, π ∫ − R 2 − r 2 R 2 − r 2 ( ( R 2 − y 2 ) 2 − r 2 ) d y = π ∫ − R 2 − r 2 R 2 − r 2 ( R 2 − y 2 − r 2 ) d y = π [ ( R 2 y − r 2 y − y 3 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 6.3, Problem 60E

To determine

The volume $V$ of the bead (Figure 17) in terms of $r$ and $R$ . Then show that $V = \frac{π}{6} h^{3}$ , where $h$ is the height of the bead and then explain intuitively, how the given statement is possible.

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Chapter 6.3, Problem 59E

Chapter 6.3, Problem 61E

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Chapter 6 Solutions

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The volume V of the bead (Figure 17) in terms of r and R . Then show that V = π 6 h 3 , where h is the height of the bead and then explain intuitively, how the given statement is possible.

Solution Summary: The author explains how the volume of the bead is given by V = 6 h 3 and how it can be explained intuitively.

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The volume $V$ of the bead (Figure 17) in terms of $r$ and $R$ . Then show that $V = \frac{π}{6} h^{3}$ , where $h$ is the height of the bead and then explain intuitively, how the given statement is possible.

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Chapter 6 Solutions

The volume V of the bead (Figure 17) in terms of r and R . Then show that V = π 6 h 3 , where h is the height of the bead and then explain intuitively, how the given statement is possible.

Solution Summary: The author explains how the volume of the bead is given by V = 6 h 3 and how it can be explained intuitively.

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The volume V of the bead (Figure 17) in terms of r and R. Then show that V=π6h3, where h is the height of the bead and then explain intuitively, how the given statement is possible.

Want to see the full answer?

Chapter 6 Solutions

The volume $V$ of the bead (Figure 17) in terms of $r$ and $R$ . Then show that $V = \frac{π}{6} h^{3}$ , where $h$ is the height of the bead and then explain intuitively, how the given statement is possible.