Chapter 5: Q5P (page 273)

Find the area of the part of the cone $z^{2} = 3 (x^{2} + y^{2})$ which is inside the sphere $x^{2} + y^{2} + z^{2} = 16$

Short Answer

Expert verified

$8 π$ Per nappe

Step by step solution

Given Condition

The equation of the cone is $z^{2} = 3 (x^{2} + y^{2})$

The equation of the sphere is $x^{2} + y^{2} + z^{2} = 16$

Concept of surface area

An intersection curve consists of the common points of two transversally intersecting surfaces. The surface area of a three-dimensional object is the total area of all its faces.

Draw the diagram

Calculate the angle

The equation of the cone is $ϕ (x, y, z) = z^{2} - 3 x^{2} - 3 y^{2}$

The secant of the angle is found as follows.

$s e c γ = \frac{\sqrt{{|\nabla ϕ|}^{2}}}{|\partial ϕ / \partial z|} = \frac{\sqrt{{|2 z \hat{z} - 6 x \hat{x} - 6 y \hat{y}|}^{2}}}{2 z} = \frac{\sqrt{z^{2} + 9 x^{2} + 9 y^{2}}}{z}$

Calculate the area.

The area of one nappe of the cone cut out by the sphere is calculated as below.

$A = \int_{0}^{2 x} d 0 \int_{0}^{2} r d r \frac{\sqrt{z^{2} + 9 r^{2}}}{z} p u t z = r c o t \frac{π}{6} = \sqrt{3} r A = 2 π \int_{0}^{2} rdr \frac{\sqrt{3 r^{2} + 9 r^{2}}}{\sqrt{3} r} = 2 π \int_{0}^{2} rdr \frac{\sqrt{12}}{\sqrt{3}} = {4 π \frac{1}{2} r^{2}}_{0}^{2} = 8 π A = 8 π per nappe .$

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Find the area of the part of the cone $z^{2} = 3 (x^{2} + y^{2})$ which is inside the sphere $x^{2} + y^{2} + z^{2} = 16$

Short Answer

Step by step solution

Given Condition

Concept of surface area

Draw the diagram

Calculate the angle

Calculate the area.

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Find the area of the part of the conez2=3(x2+y2)which is inside the spherex2+y2+z2=16

Short Answer

Step by step solution

Given Condition

Concept of surface area

Draw the diagram

Calculate the angle

Calculate the area.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Solid State Physics

Fluids

Physical Quantities And Units

Magnetism and Electromagnetic Induction

Turning Points in Physics

Study anywhere. Anytime. Across all devices.

Find the area of the part of the cone $z^{2} = 3 (x^{2} + y^{2})$ which is inside the sphere $x^{2} + y^{2} + z^{2} = 16$