Minimizing Volume with given equations and certain points. Calculus 32A

[uclagal2012]

Homework Statement

A plane with equation xa+yb+zc=1 (a,b,c>0)
together with the positive coordinate planes forms a tetrahedron of volume V=16abc (as shown in the Figure below)

Find the plane that minimizes V if the plane is constrained to pass through a point P=(8,2,3) .

Here is a picture: https://www.pic.ucla.edu/webwork2_course_files/12F-MATH32A-1/tmp/gif/dhattman-1243-setSix-prob8--image_14_8_31.png

Homework Equations

I used Lagrange multipliers SEVERAL TIMES.

The Attempt at a Solution

I've gotten
Partial(a)=(1/6)bc G(a)=-8/a^2
Partial(b)=(1/6)ac G(b)=-2/b^2
Partial(c)=(1/6)ab G(c)=-3/c^2

I then found that a=4b, 2c=3b, and 3a=8c but I am SO STUCK after this. Please help!

 

[haruspex]

tetrahedron of volume V=16abc

1/(6abc), right?

I then found that a=4b, 2c=3b, and 3a=8c

I think that's right.
So you can write each of a, b, c in terms of some common parameter, substitute that into the equation for the plane, and express the fact that the plane goes through P.

 

[Ray Vickson]

Homework Statement

A plane with equation xa+yb+zc=1 (a,b,c>0)
together with the positive coordinate planes forms a tetrahedron of volume V=16abc (as shown in the Figure below)

Find the plane that minimizes V if the plane is constrained to pass through a point P=(8,2,3) .

Here is a picture: https://www.pic.ucla.edu/webwork2_course_files/12F-MATH32A-1/tmp/gif/dhattman-1243-setSix-prob8--image_14_8_31.png

Homework Equations

I used Lagrange multipliers SEVERAL TIMES.

The Attempt at a Solution

I've gotten
Partial(a)=(1/6)bc G(a)=-8/a^2
Partial(b)=(1/6)ac G(b)=-2/b^2
Partial(c)=(1/6)ab G(c)=-3/c^2

I then found that a=4b, 2c=3b, and 3a=8c but I am SO STUCK after this. Please help!

The problem statement does not match the picture. To get the picture you need the equation of the plane to be
$$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1,$$
not what you wrote. Alternatively, you can use $ax + by + cz = 1$, but the intercepts are 1/a, 1/b and 1/c, and the volume is 1/(6*a*b*c), as stated by haruspex. So first, you need to decide which representation you want to use.

 

[uclagal2012]

The problem statement does not match the picture. To get the picture you need the equation of the plane to be
$$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1,$$
not what you wrote. Alternatively, you can use $ax + by + cz = 1$, but the intercepts are 1/a, 1/b and 1/c, and the volume is 1/(6*a*b*c), as stated by haruspex. So first, you need to decide which representation you want to use.

You are correct in that it is supposed to be
$$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1,$$

Sorry, I am not used to this forum. Additionally, the V=1/(6*a*b*c)

So now are you able to help me more?

 

[uclagal2012]

The problem statement does not match the picture. To get the picture you need the equation of the plane to be
$$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1,$$
not what you wrote. Alternatively, you can use $ax + by + cz = 1$, but the intercepts are 1/a, 1/b and 1/c, and the volume is 1/(6*a*b*c), as stated by haruspex. So first, you need to decide which representation you want to use.

You are correct in that it is supposed to be
$$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1,$$

Sorry, I am not used to this forum. Additionally, the V=(1/6)(a*b*c)

So now are you able to help me more?

 

[Ray Vickson]

You are correct in that it is supposed to be
$$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1,$$

Sorry, I am not used to this forum. Additionally, the V=(1/6)(a*b*c)

So now are you able to help me more?

You need to show your work, not just write down some unexplained equations containing undefined symbols (the G). Be explicit. What are the variables? What is the objective (the thing you are trying to maximize or minimize)? What are the constraints? (Here, I mean: write down all these objects explicitly as functions of your chosen variables.) We need to see all these things first, in order to know whether or not you are on the right track.

Assuming you have done the above, now write the Lagrangian, and then write the optimality conditions. Finally, there is the issue of solving those conditions, but again, let's see the conditions first in order to know whether you are making errors.

 

[haruspex]

You are correct in that it is supposed to be
$$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1,$$

Sorry, I am not used to this forum. Additionally, the V=(1/6)(a*b*c)

So now are you able to help me more?

Judging from the OP, you correctly obtained the relationships between a, b, and c, except that it now appears you had each inverted in the definition. So go back to those equations and switch a to 1/a etc.
Then you're almost there. All you need to is what I wrote before:
Write each of a, b, c in terms of some common parameter (t, say), substitute that into the equation for the plane, and express the fact that the plane goes through P.
What parts of that do I need to explain more?