Find the surface area of the plane x+2y+2z=12 cut off by x^2+y^2=16
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The cylinder
x2+y2=16x2+y2=16
cut off an ellipse from the plane
x+2y+2z=12x+2y+2z=12
The projection of the ellipse in the x-y plane is given by the circle:
x2+y2≤16x2+y2≤16
with radius R=4R=4 and area A=πR2=16πA=πR2=16π.
The ratio between the area of the circle and the area of the ellipse is given by the cosαcosα, αα being the angle between the normal to the plane and the zz axis.
Thus: normal to the plane: n⃗ (1,2,2)⟹cosα=23n→(1,2,2)⟹cosα=23.
The area of a quarter of ellipse is thus:
S=32⋅4π=6πS=32⋅4π=6π
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answeredDec 16 '17 at 11:59

gimusi
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First, let me know what do you think about given equations?
here, x+2y+2z=12 is a plane cutting X,Y&Z axes at 12,6,6 respectively.
x=0 is a plane i.e. YZ-plane & y=0 is XZ-plane.
x^2+y^2=16 is a circular cylinder with radius 4 and Z-axis as its axis.
Now trace all these and you will get some part of plane x+2y+2z=12 inside the cylinder b/w axes- planes. This is the portion for which to find the surface area and take projection of this portion on xy-plane , you will get 1/4 of circle of radius 4 with centre at origin.( say this circular region R)
Now, write the equation of plane in form of Monge's equation i. e. z=6-(x/2)-y and find it's partial derivatives with respect to x and y both.
formula for surface area is, S=∬ ds∬ dsover R.
here, dS =(√p2+q2+1) dxdy(p2+q2+1) dxdy;where p and q are partial derivatives of the Monge's equation with respect to x&y respectively.
p=-1/2 , q = -1 ; so ds=(3/2) dx dy
Now, S =3/2∬ dxdy∬ dxdy over R
you can see, S= (3/2)area of 1/4 of circle of radius 4 with centre at origin.
therefore S =(1/4)24ππ = 6π
x2+y2=16x2+y2=16
cut off an ellipse from the plane
x+2y+2z=12x+2y+2z=12
The projection of the ellipse in the x-y plane is given by the circle:
x2+y2≤16x2+y2≤16
with radius R=4R=4 and area A=πR2=16πA=πR2=16π.
The ratio between the area of the circle and the area of the ellipse is given by the cosαcosα, αα being the angle between the normal to the plane and the zz axis.
Thus: normal to the plane: n⃗ (1,2,2)⟹cosα=23n→(1,2,2)⟹cosα=23.
The area of a quarter of ellipse is thus:
S=32⋅4π=6πS=32⋅4π=6π
share improve this answer
answeredDec 16 '17 at 11:59

gimusi
1
up vote1down vote
First, let me know what do you think about given equations?
here, x+2y+2z=12 is a plane cutting X,Y&Z axes at 12,6,6 respectively.
x=0 is a plane i.e. YZ-plane & y=0 is XZ-plane.
x^2+y^2=16 is a circular cylinder with radius 4 and Z-axis as its axis.
Now trace all these and you will get some part of plane x+2y+2z=12 inside the cylinder b/w axes- planes. This is the portion for which to find the surface area and take projection of this portion on xy-plane , you will get 1/4 of circle of radius 4 with centre at origin.( say this circular region R)
Now, write the equation of plane in form of Monge's equation i. e. z=6-(x/2)-y and find it's partial derivatives with respect to x and y both.
formula for surface area is, S=∬ ds∬ dsover R.
here, dS =(√p2+q2+1) dxdy(p2+q2+1) dxdy;where p and q are partial derivatives of the Monge's equation with respect to x&y respectively.
p=-1/2 , q = -1 ; so ds=(3/2) dx dy
Now, S =3/2∬ dxdy∬ dxdy over R
you can see, S= (3/2)area of 1/4 of circle of radius 4 with centre at origin.
therefore S =(1/4)24ππ = 6π
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