Answer:
The dimensions are, base ![b=\sqrt[3]{200}](https://tex.z-dn.net/?f=b%3D%5Csqrt%5B3%5D%7B200%7D)
, depth ![d=\sqrt[3]{200}](https://tex.z-dn.net/?f=d%3D%5Csqrt%5B3%5D%7B200%7D)
and height ![h=\sqrt[3]{200}](https://tex.z-dn.net/?f=h%3D%5Csqrt%5B3%5D%7B200%7D)
.
Step-by-step explanation:
First we have to understand the problem, we have a box of unknown dimensions (base 
, depth 
and height 
), and we want to optimize the used material in the box. We know the volume 
we want, how we want to optimize the card used in the box we need to minimize the Area 
of the box.
The equations are then, for Volume
For Area
From the Volume equation we clear the variable to get,
And we replace this value into the Area equation to get,
So, we have our function 
, which we have to minimize. We apply the first partial derivative and equalize to zero to know the optimum point of the function, getting
After solving the system of equations, we get that the optimum point value is and , replacing this values into the equation of variable we get .
Now, we have to check with the hessian matrix if the value is a minimum,
The hessian matrix is defined as,
![H=\left[\begin{array}{ccc}\frac{\partial^2 A}{\partial d^2} &\frac{\partial^2 A}{\partial d \partial h}\\\frac{\partial^2 A}{\partial h \partial d}&\frac{\partial^2 A}{\partial p^2}\end{array}\right]](https://tex.z-dn.net/?f=H%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%5Cfrac%7B%5Cpartial%5E2%20A%7D%7B%5Cpartial%20d%5E2%7D%20%26%5Cfrac%7B%5Cpartial%5E2%20A%7D%7B%5Cpartial%20d%20%5Cpartial%20h%7D%5C%5C%5Cfrac%7B%5Cpartial%5E2%20A%7D%7B%5Cpartial%20h%20%5Cpartial%20d%7D%26%5Cfrac%7B%5Cpartial%5E2%20A%7D%7B%5Cpartial%20p%5E2%7D%5Cend%7Barray%7D%5Cright%5D)
we know that,
Then, our matrix is
![H=\left[\begin{array}{ccc}4&2\\2&4\end{array}\right]](https://tex.z-dn.net/?f=H%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D4%262%5C%5C2%264%5Cend%7Barray%7D%5Cright%5D)
Now, we found the eigenvalues of the matrix as follow
![det(H-\lambda I)=det(\left[\begin{array}{ccc}4-\lambda&2\\2&4-\lambda\end{array}\right] )=(4-\lambda)^2-4=0](https://tex.z-dn.net/?f=det%28H-%5Clambda%20I%29%3Ddet%28%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D4-%5Clambda%262%5C%5C2%264-%5Clambda%5Cend%7Barray%7D%5Cright%5D%20%29%3D%284-%5Clambda%29%5E2-4%3D0)
Solving for
, we get that the eigenvalues are: 
and 
, how both are positive the Hessian matrix is positive definite which means that the function
is minimum at that point.
