Chapter 5: Q1P (page 272)

Find the area of the plane $x - 2 y + 5 z = 13$ cut out by the cylinder $x^{2} + y^{2} = 9$ .

Short Answer

Expert verified

The area of the plane $x - 2 y + 5 z = 13$ cut out by the cylinder $x^{2} + y^{2} = 9$ is $A = 9 π \frac{\sqrt{30}}{5} .$

Step by step solution

Given Condition

The equation of the plane is $x - 2 y + 5 z = 13$ and the equation of the cylinder is $x^{2} + y^{2} = 9$ .

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 secant of the angle $γ$ is given by,

$s e c γ = \frac{\sqrt{{|\nabla ϕ|}^{2}}}{|\partial ϕ / \partial z|} = \frac{\sqrt{{|\hat{x} - 2 \hat{y} + 5 \hat{z}|}^{2}}}{5} = \frac{\sqrt{1 + 4 + 5}}{5} = \frac{\sqrt{30}}{5}$

Calculate the area

The area of the plane cut by the cylinder can be found by,

$A = \int_{x} \int_{y} s e c γ d x d y = \frac{\sqrt{30}}{5} (9 π) = 9 π \frac{\sqrt{30}}{5}$

Hence, the required area is $A = 9 π \frac{\sqrt{30}}{5}$ .

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Find the area of the plane $x - 2 y + 5 z = 13$ cut out by the cylinder $x^{2} + y^{2} = 9$ .

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 plane x-2y+5z=13 cut out by the cylinder x2+y2=9.

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

Nuclear Physics

Electricity

Atoms and Radioactivity

Force

Classical Mechanics

Mechanics and Materials

Study anywhere. Anytime. Across all devices.

Find the area of the plane $x - 2 y + 5 z = 13$ cut out by the cylinder $x^{2} + y^{2} = 9$ .