Find the area of the part of the plane 5x + 3y + z = 15 that lies in the first octant.

Lewis Harvey

Lewis Harvey

Answered question

2021-05-16

Find the area of the part of the plane 5x+3y+z=15 that lies in the first octant.

Answer & Explanation

au4gsf

au4gsf

Skilled2021-05-17Added 95 answers

I used the following method:

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Jeffrey Jordon

Jeffrey Jordon

Expert2021-09-09Added 2605 answers

Find the area of the part of the plane.

5x+3y+2=15

that lies in the first octant parametrization:

R=<x,y,z>

=<x,y,155x3y>

Rx=<1,0,5>

and Ry=<0,1,3>

so, Rx×Ry=[ijk105013]=<5,3,1>

|Rx×Ry|=52+32+12=25+9+1=35

to see where the plane intersects the first octant, look for the intercepts with the x,y and z-axes

5x+3y+2=15

5x15+3y15+215=1

x3+y5+215=1

So, (3,0,0), (0,5,0), (0,0,15)

thus, the domain D is the triangle in the first quadrant of the xy-plane bounded by (0,0),(3,0) and (0,5).

We could compute this using geometry area =D35dA

=35DdA=35 area of triangle

=35(12×3×5)=15235

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