Answered: A decorative oak post is 30 inches long…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

$A decorative oak post is 30 inches long and is turned on a lathe so that its profile is sinusoidal as shown in the figure below. The figure depicts a 3D model of the oak post, which is sinusoidal with varying radii along its length. The model shows a symmetrical vase-like shape with wider sections interspersed by narrower waists. Annotations indicate the initial radius $ r_0 $ at the top and a distance $ a_0 $ from the top to the point where the radius is minimum. In this figure, $ r_0 = 4 $ inches and $ a_0 = 10 $ inches. (a) Describe the surface of the post parametrically using cylindrical coordinates and the parameters $ s $ and $ t $. \[ x(s, t) = \left(1 + \cos\left(\frac{\pi s}{5}\right)\right) \cos(t), \] \[ y(s, t) = \left(1 + \cos\left(\frac{\pi s}{5}\right)\right) \sin(t), \] \[ z(s, t) = s, \quad \text{where} \quad 0 \leq s \leq 30 \quad \text{and} \quad 0 \leq t \leq 2\pi. \] (b) Find the volume of the post. \[ \text{volume} = \,\, ? \] (Include $\text{units.}$)$

Transcribed Image Text:A decorative oak post is 30 inches long and is turned on a lathe so that its profile is sinusoidal as shown in the figure below. The figure depicts a 3D model of the oak post, which is sinusoidal with varying radii along its length. The model shows a symmetrical vase-like shape with wider sections interspersed by narrower waists. Annotations indicate the initial radius $ r_0 $ at the top and a distance $ a_0 $ from the top to the point where the radius is minimum. In this figure, $ r_0 = 4 $ inches and $ a_0 = 10 $ inches. (a) Describe the surface of the post parametrically using cylindrical coordinates and the parameters $ s $ and $ t $. \[ x(s, t) = \left(1 + \cos\left(\frac{\pi s}{5}\right)\right) \cos(t), \] \[ y(s, t) = \left(1 + \cos\left(\frac{\pi s}{5}\right)\right) \sin(t), \] \[ z(s, t) = s, \quad \text{where} \quad 0 \leq s \leq 30 \quad \text{and} \quad 0 \leq t \leq 2\pi. \] (b) Find the volume of the post. \[ \text{volume} = \,\, ? \] (Include $\text{units.}$)

Expert Solution