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Thread: Maximize Volume with constraint

  1. #1
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    Maximize Volume with constraint

    Hello all, I am stuck on a Calc. 3 problem involving maximization. I've got the word problem into equations but I get stuck finding the gradient of the constraint function. We are supposed to be using Lagrange Multipliers to solve.

    Problem: The material for constructing the base of an open box costs 1.5 times as much per unit are that of it's sides. For a fixed amount of money C, find the dimensions of the box with the largest volume that can be made.

    From that I got:
    V(x,y,z) = xyz constrained to: 3xy + 4xz + 4yz = 2C

    So $\displaystyle \nabla \$V(x,y,z) = yzi + xzj + yzk

    But what is the gradient of the constraining function? Any help is appreciated!
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  2. #2
    Super Member Aryth's Avatar
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    Remember when dealing with a constraint function you have to remove the constants and create a new function g(x,y) (basically, the entire left hand side of the equation) that consists of all the variable terms of the function, giving you:

    $\displaystyle g(x,y) = 3xy + 4xz + 4yz$

    Making your gradient:

    $\displaystyle \nabla g(x,y) = (3y + 4z)\hat{i} + (3x + 4z)\hat{j} + (4x + 4y)\hat{k}$

    You know that:

    $\displaystyle \nabla f(x,y) = \lambda \nabla g(x,y)$

    Which gives you a system of equations:

    $\displaystyle f_x = \lambda g_x$
    $\displaystyle f_y = \lambda g_y$
    $\displaystyle f_z = \lambda g_z$
    3xy + 4xz + 4yz = 2C

    Solve for lambda first, then get equations in terms of JUST x,y, and z, and solve for each simultaneously (Unless solving for lambda gives you some clue as to the solution of 1 or more variables) and you will get the critical points.
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  3. #3
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    I had forgotten about making g(x,y)

    So setting $\displaystyle \nabla f(x,y) = \lambda \nabla g(x,y)$ would look like...

    $\displaystyle yz = 7 \lambda$
    $\displaystyle xz = 7 \lambda$
    $\displaystyle xy = 8 \lambda$

    which gives me

    $\displaystyle xz=yz \to x=y$ but how to I incorporate z into my system of equations?
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  4. #4
    Super Member Aryth's Avatar
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    Quote Originally Posted by CleanSanchez View Post
    I had forgotten about making g(x,y)

    So setting $\displaystyle \nabla f(x,y) = \lambda \nabla g(x,y)$ would look like...

    $\displaystyle yz = 7 \lambda$
    $\displaystyle xz = 7 \lambda$
    $\displaystyle xy = 8 \lambda$

    which gives me

    $\displaystyle xz=yz \to x=y$ but how to I incorporate z into my system of equations?
    Your system is a little off. You are adding the numbers, but you are not taking the variables into account.

    Notice that $\displaystyle g_x = 3y + 4z$

    Which means:

    $\displaystyle f_x = \lambda g_x$

    gives you:

    $\displaystyle yz = \lambda (3y + 4z)$

    So, doing the same thing for the others, your system of equations becomes:

    $\displaystyle yz = \lambda (3y + 4z)$
    $\displaystyle xz = \lambda (3x + 4z)$
    $\displaystyle xy = \lambda (4x + 4y)$
    $\displaystyle 3xy + 4xz + 4yz = 2C$

    Now, take the first one and solve for lambda:

    $\displaystyle \lambda = \frac{yz}{(3y + 4z)}$

    This is not a pretty lambda, so try taking the first one and solving for it:

    $\displaystyle \lambda = \frac{xz}{(3x + 4z)}$

    Now, lets set these equal to each other, since lambda doesn't change:

    $\displaystyle \frac{yz}{(3y + 4z)} = \frac{xz}{(3x + 4z)}$

    This is a proportion, and you can make this:

    $\displaystyle yz(3x + 4z) = xz(3y + 4z)$

    Take a z out:

    $\displaystyle y(3x + 4z) = x(3y + 4z)$

    Now, we distribute on both sides:

    $\displaystyle 3xy + 4yz = 3xy + 4xz$

    We now get that:

    $\displaystyle 4yz = 4xz$

    This simplifies to:

    $\displaystyle y = x$, this is the result you got, but using the correct method.

    Now, go back to your system of equations and pick one with with just x and y to find an actual lambda, the third one qualifies:

    $\displaystyle xy = \lambda (4x + 4y)$

    $\displaystyle xy = (8y) \lambda$

    $\displaystyle \frac{x}{8} = \lambda$

    Now, we find one with x's and z's, the second one qualifies:

    $\displaystyle xz = \frac{x}{8} (3x + 4z)$

    $\displaystyle z = \frac{(3x + 4z)}{8}$

    $\displaystyle z - \frac{1}{2}z = \frac{3}{8}x$

    $\displaystyle \frac{1}{2}z = \frac{3}{8}x$

    $\displaystyle x = \frac{4}{3}z = y$

    This gives you an interesting result:

    $\displaystyle 3x = 4z = 3y$

    Now, we can reorganize the last equation from:

    $\displaystyle 3xy + 4xz + 4yz = 2C$

    to:

    $\displaystyle (4z)y + (4z)x + (4z)y = 2C$

    We know that x = y, so:

    $\displaystyle 12yz = 2C$

    $\displaystyle 6yz = C$

    $\displaystyle yz = \frac{C}{6} = xz$

    Now, we can go back to the first equation and sub in for yz:

    $\displaystyle \frac{C}{6} = \frac{x}{8}(3y + 4z)$

    We know that 4z = 3y, so:

    $\displaystyle \frac{C}{6} = 6y\frac{x}{8}$

    We also know that x = y

    $\displaystyle \frac{C}{6} = y^2\frac{3}{4}$

    $\displaystyle \frac{2C}{3} = 3y^2$

    $\displaystyle \frac{2C}{9} = y^2$

    $\displaystyle \frac{\sqrt{2C}}{3} = y = x$

    We plug this into the altered equation (6yz = C) and solve for z

    $\displaystyle 6\left(\frac{\sqrt{2C}}{3}\right)z = C$

    $\displaystyle 2\sqrt{2C}z = C$

    $\displaystyle z = \frac{\sqrt{C}}{2\sqrt{2}} = \frac{\sqrt{2C}}{4}$

    So, we have the critical points:

    $\displaystyle \left(\frac{\sqrt{2C}}{3}, \frac{\sqrt{2C}}{3}, \frac{\sqrt{2C}}{4}\right)$

    and

    $\displaystyle \left(-\frac{\sqrt{2C}}{3}, -\frac{\sqrt{2C}}{3}, -\frac{\sqrt{2C}}{4}\right)$

    Because of the radicals, and if x is positive, then y and z must be positive, and if x is negative, then y and z must be negative, so any other mixture of signs doesn't follow the equations. But notice, these two points will yield the same result, so by default, it is the maximum.

    I'm hoping these are right, if they're not, I missed something within all the algebra and you'll probably be able to figure it out on your own. Good luck.
    Last edited by Aryth; Mar 26th 2009 at 04:12 PM.
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  5. #5
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    Thanks so much, you're a life saver! I'm pretty sure this is the hardest problem my professor will assign because the system was much more difficult than any of the other examples. Thanks again, this website seems like an excellent resource!
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  6. #6
    Super Member Aryth's Avatar
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    It's no problem. And that is a difficult example, I have no idea why a teacher would put people through that without even giving the some kind of head start.
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