Can anyone help me solve the problem below? This is question number 14.8.42 in the seventh edition of Stewart Calculus.
Here is the problem definition:
"Find the maximum and minimum volumes of a rectangular box whose surface area is $1500 cm^3$ and whose total edge length is 200 cm."
Here is my work so far(With edits based on Phyra's suggestions below):
$V=f(x,y,z)=xyz$
Subject to two constraints:
Surface area: $g(x,y,z)=2xy+2xz+2yz=1500$
Edge length: $h(x,y,z)=4x+4y+4z=200$
Simplify constraints to:
$g(x,y,z)=xy+xz+yz=750$
$h(x,y,z)=x+y+z=50$
Solve system of equations:
$f_x=yz=\lambda (y+z)+\mu$
$f_y=xz=\lambda (x+z)+\mu$
$f_z=xy=\lambda (x+y)+\mu$
This leads to:
$\lambda y+\lambda z+\mu=yz$
$\lambda x+\lambda z+\mu=xz$
$\lambda x+\lambda y+\mu=xy$
Thus,
$\lambda x=xy-\lambda y-\mu=xz-\lambda z-\mu$, so that $xy-\lambda y=xz-\lambda z$, which simplifies to: $x(y-z)=\lambda (y-z)$, so that $\lambda=x$ when $(y-z)\ne 0$
Similarly,
$\lambda y =yz-\lambda z -\mu=xy-\lambda x -\mu$, so that $y(z-x)=\lambda (z-x)$, and $\lambda = y$ when $(z-x)\ne 0$
Also,
$\lambda z = yz-\lambda y -\mu=xz-\lambda x - \mu$, so that $z(y-x)=\lambda (y-x)$, and $\lambda = z$ when $(y-x)\ne 0$
The above can be summarized as:
$\lambda=x=y=z $
Substitute x=y into constraints:
$x^2 +2xz =750$ and $2x+z=50$
Substitute $z=50-2x$ into first constraint:
$3x^2-100x+750=0$ gives $x=y=\frac{5}{3}(\frac{+}{}\sqrt{10}+10)$
Also, $z=50-2x=\frac{50\frac{+}{}10\sqrt{10}}{3}$
Thus, $V=xyz\approx 2948$ or $\approx 3534$
My remaining questions are as follows:
a) What do you mean in your comments below when you refer to checking the border? This concept is not in my textbook.
b) Why choose x=y? Are you saying that choosing y=z would have given the same result?
c) Why use $z=50-2x$ instead of $x=y=z$?
d) When you mention 8 cases, to what are you referring? I do not see any cases.
e) What do you mean when you use the word symmetric in this situation?