Question

Surface area of plane
plane x + 2y + 2 Z= 10cut off
by x = 0, y = 0 = 1, y = 1
+ 2y + 2z= 10 ​

Answer 1

Answer:

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 cosαcos⁡α, 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)⟹cos⁡α=23.

The area of a quarter of ellipse is thus:

S=32⋅4π=6π