Surface portion bounded by plane 2x +5y + z = 10 that lies

[chetzread]

Homework Statement

Find the surface portion bounded by plane 2x +5y + z = 10 that lies in cylinder (x^2) +(y^2) = 9 ...

I have skteched out the diagram and my ans is 5sqrt(30) instead of 9sqrt (30) as given by the author ...
Anything wrong with my working ?

Homework Equations

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

Find the surface portion bounded by plane 2x +5y + z = 10 that lies in cylinder (x^2) +(y^2) = 9 ...

I have skteched out the diagram and my ans is 5sqrt(30) instead of 9sqrt (30) as given by the author ...
Anything wrong with my working ?

Homework Equations

The Attempt at a Solution

Your handwriting is so hard to read that I will not even try. Type out your work, as is the PF standard. (Read the post "Guidelines for students and helpers", by Vela.)

 

[chetzread]

Your handwriting is so hard to read that I will not even try. Type out your work, as is the PF standard. (Read the post "Guidelines for students and helpers", by Vela.)

Which part you can't see it clearly?

 

[Mark44]

Regarding your answer, I don't see that you have used the given information that the portion of the plane you want is inside the cylindrical cylinder.

Which part you can't see it clearly?

The part between the top and bottom of what you wrote. I agree with Ray that your hand-written work is hard to read, and we strongly discourage posting the work in an image. Except for the sketches of the surface and the triangle, everything can be done using LaTeX. For a tutorial, see https://www.physicsforums.com/help/latexhelp/ (under the INFO menu, in the Help/How-to submenu)
Here's a quick lesson:
Fractions:
$\frac{a + b}{c}$ -- LaTeX script ##\frac{a + b}{c}##
Square roots: $\sqrt{30}$ -- LaTeX script ##\sqrt{30}##
Integrals $\int_{z = 0}^{2} f(z) dz$ -- LaTeX script ##\int_{z = 0}^{2} f(z) dz##
$\int_{x = 0}^{10}\int_{y = 5}^{x} f(x, y) dy dx$ -- LaTeX script ##\int_{x = 0}^{10}\int_{y = 5}^{x} f(x, y) dy dx##

 

[LCKurtz]

Regarding your answer, I don't see that you have used the given information that the portion of the plane you want is inside the cylindrical cylinder.

The part between the top and bottom of what you wrote. I agree with Ray that your hand-written work is hard to read, and we strongly discourage posting the work in an image. Except for the sketches of the surface and the triangle, everything can be done using LaTeX. For a tutorial, see https://www.physicsforums.com/help/latexhelp/ (under the INFO menu, in the Help/How-to submenu)
Here's a quick lesson:
Fractions:
$\frac{a + b}{c}$ -- LaTeX script ##\frac{a + b}{c}##
Square roots: $\sqrt{30}$ -- LaTeX script ##\sqrt{30}##
Integrals $\int_{z = 0}^{2} f(z) dz$ -- LaTeX script ##\int_{z = 0}^{2} f(z) dz##
$\int_{x = 0}^{10}\int_{y = 5}^{x} f(x, y) dy dx$ -- LaTeX script ##\int_{x = 0}^{10}\int_{y = 5}^{x} f(x, y) dy dx##

@Mark44 What's with that LaTeX script thing? I like it and have had many occasions where I would have liked to use it, but it doesn't work for me, even when I copy yours as below. It just renders it:

LaTeX script $2^2$
LaTeX script $\int_{z = 0}^{2} f(z) dz$

 

[Mark44]

@Mark44 What's with that LaTeX script thing? I like it and have had many occasions where I would have liked to use it, but it doesn't work for me, even when I copy yours as below. It just renders it:

LaTeX script $2^2$
LaTeX script $\int_{z = 0}^{2} f(z) dz$

I did something tricky in the part where I show the unrendered LaTeX: In the first # sign of the start and end pair, I change the text color, using the black and white circle (the fourth icon from the left on the menu bar. It doesn't matter what color you choose -- the presence of a bbcode color tag keeps the browser from rendering the LaTeX script as it normally would.

 

[chetzread]

$\int_{y = 0}^{2} \int_{z=0}^{10-5y} \sqrt {(-0.5)^2 +(-2.5)^2 +1} dzdy$[/QUOTE]

$\int_{y = 0}^{2} \int_{z=0}^{10-5y} \ 0.5 sqrt {30} dzdy$

$0.5 sqrt {30}\int_{y = 0}^{2} \ dy$
=
$5sqrt {30}$

@Mark44 , here's my working , i am not sure whether my projection to zy plane is correct or not ...

 

[chetzread]

anyone else can help ?

 

[LCKurtz]

$\int_{y = 0}^{2} \int_{z=0}^{10-5y} \sqrt {(-0.5)^2 +(-2.5)^2 +1} dzdy$

$\int_{y = 0}^{2} \int_{z=0}^{10-5y} \ 0.5 sqrt {30} dzdy$

$0.5 sqrt {30}\int_{y = 0}^{2} \ dy$
=
$5sqrt {30}$

@Mark44 , here's my working , i am not sure whether my projection to zy plane is correct or not ...

I have a couple of comments for you. First: your statement of the problem says the area of the plane inside the cylinder, but your picture and working apparently are for only the portion of the surface in the first octant only. Which is it?

Second: The intersection of a plane and a cylinder is typically an ellipse. Why on Earth would you choose the projection on the zy plane? The only projection that would be circular would be on the xy plane. That's where you should project it.

 

[chetzread]

but your picture and working apparently are for only the portion of the surface in the first octant only. Which is it?

Sorry , i forgot to include other octant ...

Second: The intersection of a plane and a cylinder is typically an ellipse. Why on Earth would you choose the projection on the zy plane? The only projection that would be circular would be on the xy plane. That's where you should project it.

why the projection to xy plane is a circle ? why shouldn't the projection of xy plane is the area of the plane that lies inside the cylinder ? refer to the green part ... that's the part of plane

 

[LCKurtz]

Sorry , i forgot to include other octant ...

why the projection to xy plane is a circle ? why shouldn't the projection of xy plane is the area of the plane that lies inside the cylinder ? refer to the green part ... that's the part of plane

No it isn't. The boundary of a plane cutting through a cylinder at an angle will be curved and, in this case, is an ellipse shape.

 

[chetzread]

No it isn't. The boundary of a plane cutting through a cylinder at an angle will be curved and, in this case, is an ellipse shape.

why ? can you explain further with the aid of diagram ? I still can't imagine it ...Btw , the plane 2x + 5y + z = 10 only lies in first octant , am i right ?

 

[LCKurtz]

why ? can you explain further with the aid of diagram ? I still can't imagine it ...Btw , the plane 2x + 5y + z = 10 only lies in first octant , am i right ?

A plane extends in all directions. Your triangle is just a small portion of a plane. Forget the coordinate system for a minute. Just imagine a cylinder with a plane slicing through it. Here's a picture of one (not your particular one). There aren't any straight lines like you have drawn. Also, although the green is an ellipse, do you see that if you looked at it straight down the z axis it would look like a circle?

 

[chetzread]

A plane extends in all directions. Your triangle is just a small portion of a plane. Forget the coordinate system for a minute. Just imagine a cylinder with a plane slicing through it. Here's a picture of one (not your particular one). There aren't any straight lines like you have drawn. Also, although the green is an ellipse, do you see that if you looked at it straight down the z axis it would look like a circle?

ok , i can undetstand that ... But , i still couldn't imagine that why my case is same as above (the cylinder that is being sliced above) ... The plane 2x+5y +z =10 is a triangular plane , right ? It's not the top part of cylinder that is being sliced

 

[LCKurtz]

ok , i can undetstand that ... But , i still couldn't imagine that why my case is same as above (the cylinder that is being sliced above) ... The plane 2x+5y +z =10 is a triangular plane , right ? It's not the top part of cylinder that is being sliced

Here's what your original problem statement was:

Homework Statement

Find the surface portion bounded by plane 2x +5y + z = 10 that lies in cylinder (x^2) +(y^2) = 9 ...

There is nothing about triangles, octants, or straight lines. When planes cut through cylinders at an angle it looks generally like the picture I gave you.

 

[chetzread]

Here's what your original problem statement was:
There is nothing about triangles, octants, or straight lines. When planes cut through cylinders at an angle it looks generally like the picture I gave you.

why ? The 2x+5y +z =10 is a triangular plane... The diagram that you sketched look like the volume bounded by tirangular plane 2x+5y +z =10 and cylinder ... But , what we want here is the surface area of plane 2x+5y +z =10 inside cylinder , right ? So , how could the surface area of tirangular plane 2x+5y +z =10 look like a circular plane ?

 

[LCKurtz]

I can't tell whether you are trolling me or are just hopeless. Either way it's pretty clear I can't get through to you with this problem so I am abandoning this thread. Maybe someone else will continue.

 

[Mark44]

The 2x+5y +z =10 is a triangular plane...

No!
A plane is not triangular nor is it circular, as you say below.

The diagram that you sketched look like the volume bounded by tirangular plane 2x+5y +z =10 and cylinder

The diagram that LCKurtz provided shows the surface area (not volume) of the region of a plane inside a cylinder, very similar to what you're trying to find in this problem.

Until you can understand why the diagram that you drew is incorrect, and the one that LCKurtz provided is what you need, you will not be able to do this problem.

... But , what we want here is the surface area of plane 2x+5y +z =10 inside cylinder , right ? So , how could the surface area of tirangular plane 2x+5y +z =10 look like a circular plane ?

Again, your statements about a "triangular plane" and a "circular plane" make no sense.

 

[chetzread]

No!
A plane is not triangular nor is it circular, as you say below.
The diagram that LCKurtz provided shows the surface area (not volume) of the region of a plane inside a cylinder, very similar to what you're trying to find in this problem.

Until you can understand why the diagram that you drew is incorrect, and the one that LCKurtz provided is what you need, you will not be able to do this problem.

Again, your statements about a "triangular plane" and a "circular plane" make no sense.

if we look at the plane 2x+5y +z =10 only , then it's triangular plane right ?

 

[Mark44]

if we look at the 2x+5y +z =10 only , then it's triangular plane right ?

No! If you look at only the portion of the plane in the first octant, what you get is a triangle, but the plane extends infinitely far in two dimensions.

BTW, there is no such thing as a "triangular plane" or a "circular plane."

 

[chetzread]

No! If you look at only the portion of the plane in the first octant, what you get is a triangle, but the plane extends infinitely far in two dimensions.

BTW, there is no such thing as a "triangular plane" or a "circular plane."

well , in the previous example , i can understand that plane z= x +4 lies in the cylinder has the surface area as shown... I just don't understand why that the plane 2x+5y +z = 10 has the shape similar to the case of z= x +4 lies in the cylinder as shown...

IMO , the y -intercept is 2 , which is less than the y-intercept of cylinder , which is 3 , so , how could the surface area formed by 2x+5y +z = 10 inside the cylinder has circular surface ?

 

[Mark44]

well , in the previous example , i can understand that plane z= x +4 lies in the cylinder has the surface area as shown... I just don't understand why that the plane 2x+5y +z = 10 has the shape similar to the case of z= x +4 lies in the cylinder as shown...

IMO , the y -intercept is 2 , which is less than the y-intercept of cylinder , which is 3 , so , how could the surface area formed by 2x+5y +z = 10 inside the cylinder has circular surface ?

The plane doesn't stop at the x-y plane, as you seem to be thinking. The portion of the plane that is inside the circular cylinder has an elliptical shape, a point that has been made multiple times in this thread.

Look at it this way: if the plane were parallel to the x-y plane, the portion of this plane that is inside the cylinder would be circular, right? If you tilt the plane, as 2x + 5y + z = 10 is, the portion of this plane that is inside the cylinder is in the shape of an ellipse.

You're stuck on the idea that you plane has a triangular shape. IT DOESN'T!

 

[chetzread]

If you tilt the plane, as 2x + 5y + z = 10 is, the portion of this plane that is inside the cylinder is in the shape of an ellipse.

how to tilt the plane ? which angle and which direction should i tilt it ?

 

[Mark44]

how to tilt the plane ? which angle and which direction should i tilt it ?

Tilt it in your mind. Pretty much any angle is fine, but don't tilt it 90° or any multiple of 90°.

 

[chetzread]

Tilt it in your mind. Pretty much any angle is fine, but don't tilt it 90° or any multiple of 90°.

do you mean if we extend the plane 2x+5y +z = 10 infinitely beyond xyz plane , then the plane it's a very large triangle plane ...So , the part where the triangle plane intersecting the cylinder would be the circular part (ellipse ) only ? just like what i did below ?

 

[Mark44]

do you mean if we extend the plane 2x+5y +z = 10 infinitely beyond xyz plane , then the plane it's a very large triangle plane ...So , the part where the triangle plane intersecting the cylinder would be the circular part (ellipse ) only ? just like what i did below ?

Your drawing of the intersection of the plane and cylinder is not too far off.

Some of the other things you said are incorrect, though.
"if we extend the plane 2x+5y +z = 10 infinitely beyond xyz plane" - there is no "xyz" plane. There are the three coordinate planes: x-y plane, x-z plane, and y-z plane.

"then the plane it's a very large triangle plane" -- there's no such thing as a "triangle plane"! The plane doesn't stop at the edges of the triangle -- the plane continues on indefinitely.