Q10P a) Find the volume inside the co... [FREE SOLUTION]

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Chapter 5: Q10P (page 268)

a) Find the volume inside the cone $3 z^{2} = x^{2} + y$ , above the plane $z = 2$ and inside the sphere $x^{2} + y^{2} + z^{2} = 36$ . Hint: Use spherical coordinates.
b) Find the centroid of the volume in (a)

Short Answer

Expert verified

(a). The required volume is $64 蟿蟿$ .

(b). The centroid is $(0, 0, \frac{231}{64})$ .

Step by step solution

Given information

The solid is bounded by the cone $3 z^{2} = x^{2} + y$ , the plane $z = 2$ and the sphere $x^{2} + y^{2} + z^{2} = 36$ .

Concept of spherical coordinates

The spherical coordinates $(r, 胃, 蠒)$ is related to Cartesian coordinates (x,y,z)by:

$r = \sqrt{x^{2} + y^{2} + z^{2} 胃} = \tan^{- 1} (\frac{y}{x}) 蠒 = \cos^{- 1} (\frac{z}{r})$

The cylindrical coordinates $(r, 胃, 蠒)$ is related to Cartesian coordinates(x,y,z) by:

$r = \sqrt{x^{2} + y^{2} 胃} = t a n^{- 1} (\frac{y}{x}) z = z$

Substitute 0 for in the equation of cone.

(a)

Calculate the volume of cone as follows:

$z^{2} = \frac{x^{2}}{3} z = \frac{x^{2}}{\sqrt{3}} 伪 = \arctan (\frac{x}{z}) = \frac{蟿蟿}{3}$

The required volume is as follows:

$V = {鈭�}_{\frac{2}{\cos 胃}}^{6} r^{2} d r {鈭�}_{0}^{蟿蟿 / 3} \sin 胃 d 胃 {鈭�}_{0}^{2 蟿蟿} d 蠒 = \frac{2 蟿蟿}{3} {[r^{3}]}_{\frac{2}{\cos 胃}}^{0} {鈭�}_{0}^{蟿蟿 / 3} \sin 胃 {[r^{3}]}_{\frac{2}{\cos 胃}}^{6} d 胃 = \frac{2 蟿蟿}{3} {(108 - \frac{4}{\cos^{2} 胃})}_{0}^{蟿蟿 / 3} = 64 蟿蟿$

Thus, the required volume is $64 蟿蟿$ .

Use rcosθ for z in the formula for centroid

(b)

Here, $x = y = 0 z = \frac{1}{V} {鈭�}_{v} z d V = \frac{1}{64 蟿蟿} {鈭�}_{\frac{2}{\cos 胃}} r^{3} d r {鈭�}_{0}^{蟿蟿 / 3} \sin 胃 \cos 胃 d 胃 {鈭�}_{0}^{2 蟿蟿} d 蠒 = \frac{1}{128} {[- 324 \cos (2 胃) - \frac{8}{\cos^{2} 胃}]}_{0}^{蟿蟿 / 3} = \frac{231}{64}$ .

Thus, the centroid is $(0, 0, \frac{231}{64})$ .

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91影视

a) Find the volume inside the cone $3 z^{2} = x^{2} + y$ , above the plane $z = 2$ and inside the sphere $x^{2} + y^{2} + z^{2} = 36$ . Hint: Use spherical coordinates.
b) Find the centroid of the volume in (a)

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

Substitute 0 for in the equation of cone.

Use rcosθ for z in the formula for centroid

One App. One Place for Learning.

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91影视

a) Find the volume inside the cone3z2=x2+y, above the planez=2 and inside the sphere x2+y2+z2=36. Hint: Use spherical coordinates.b) Find the centroid of the volume in (a)

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

Substitute 0 for in the equation of cone.

Use rcosθ for z in the formula for centroid

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fluids

Radiation

Scientific Method Physics

Kinematics Physics

Torque and Rotational Motion

Fields in Physics

Study anywhere. Anytime. Across all devices.

a) Find the volume inside the cone $3 z^{2} = x^{2} + y$ , above the plane $z = 2$ and inside the sphere $x^{2} + y^{2} + z^{2} = 36$ . Hint: Use spherical coordinates.
b) Find the centroid of the volume in (a)