Find an example for 2 planes with parameters

[gipc]

I have to find two planes
z=f_1(x,y)=a1*x+b1*y+c1
z=f_2(x,y)=a2*x+b2*y+c2

That satisfy:
1. both planes go through (-5,9,8)
2. the intersection line between the planes is located outside of the Cylinder x^2+y^2=4
3. the surface area of x^2+y^2=4 that is bounded between f1 and f2 is exactly 11*pi

 

[hunt_mat]

For 1) insert the point (-5,9,8) in both equation, you will be able to eliminate one of the variables in each of the equations this way.
For 2) Compute the intersection of the line and use the condition to help you get rid of some more.
For 3) A surfce integration is required.

Looks like a fun question!

 

[gipc]

Not fun nor funny.

I don't master Surface Integration very well.

What I've found so far:
Plane 1: z = 8

Which leaves you with a circle on the bottom, which has an area of 4π, an ellipse on the top, and a cylindrical strip along the outside.

Now I have to just take a surface area integral of the function, setting a2 = 0, so you only have to solve for b2, and set that area equal to a 11π.

The problem is I'm not sure on the technique that follows in the Integral :(

 

[hunt_mat]

Doing part 1 shows that:
<br /> \begin{array}{ccc}<br /> 8 &amp; = &amp; -5b_{1}+9b_{1}+c_{1} \\<br /> 8 &amp; = &amp; -5b_{2}+9b_{2}+c_{2}<br /> \end{array}<br />
Getting rid of c_{1} and c_{2} shows that:
<br /> \begin{array}{ccc}<br /> z &amp; = &amp; a_{1}(x+5)+b_{1}(y-9)+8 \\<br /> z &amp; = &amp; a_{2}(x+5)+b_{2}(y-9)+8<br /> \end{array}<br />
Now compute the line at which the intersect.

 

[LCKurtz]

I haven't worked through your problem but here's a hint that might help you. Think of a cylinder sitting on the xy plane at the origin and a slanted plane passing through (0,0,h) cutting the cylinder. Draw a picture and also include the horizontal plane cutting the cylinder at z = h. The slanted plane and the horizontal plane form a couple of wedges.

Do you see any symmetry you can use?