Question

A company wishes to manufacture some boxes out of card. The boxes will have 6 sides (i.e. they covered at the top). They wish the boxes to have a volume of 200 cm3. What dimensions should they make the boxes in order to minimise the amount of card used? You can assume no connect the sides of the box. You should use the Hessian matrix to prove this is indeed a minimum.. [10 marks are flaps required to are

Answer 1

The dimensions are, base
`b=\sqrt[3]{200}`, depth
`d=\sqrt[3]{200}` and height
`h=\sqrt[3]{200}`.

Explanation:

First we have to understand the problem, we have a box of unknown dimensions (base
`b`, depth
`d` and height
`h`), and we want to optimize the used material in the box. We know the volume
`V` we want, how we want to optimize the card used in the box we need to minimize the Area
`A` of the box.

The equations are then, for Volume

`V=200cm^3 = b.h.d`

For Area

`A=2.b.h+2.d.h+2.b.d`

From the Volume equation we clear the variable
`b` to get,

`b=(200)/(d.h)`

And we replace this value into the Area equation to get,

`A=2.((200)/(d.h) ).h+2.d.h+2.((200)/(d.h) ).d`

`A=2.((200)/(d) )+2.d.h+2.((200)/(h) )`

So, we have our function
`f(x,y)=A(d,h)`, which we have to minimize. We apply the first partial derivative and equalize to zero to know the optimum point of the function, getting

`(\partial A)/(\partial d) =-(400)/(d^2)+2h=0`

`(\partial A)/(\partial h) =-(400)/(h^2)+2d=0`

After solving the system of equations, we get that the optimum point value is
`d=\sqrt[3]{200}` and
`h=\sqrt[3]{200}`, replacing this values into the equation of variable
`b` we get
`b=\sqrt[3]{200}`.

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}(\partial^2 A)/(\partial d^2) &(\partial^2 A)/(\partial d \partial h)\\(\partial^2 A)/(\partial h \partial d)&(\partial^2 A)/(\partial p^2)\end{array}\right]`

we know that,

`(\partial^2 A)/(\partial d^2)=(\partial)/(\partial d)(-(400)/(d^2)+2h )=(800)/(d^3)`

`(\partial^2 A)/(\partial h^2)=(\partial)/(\partial h)(-(400)/(h^2)+2d )=(800)/(h^3)`

`(\partial^2 A)/(\partial d \partial h)=(\partial^2 A)/(\partial h \partial d)=(\partial)/(\partial h)(-(400)/(d^2)+2h )=2`

Then, our matrix is

`H=\left[\begin{array}{ccc}4&2\\2&4\end{array}\right]`

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`

Solving for
`\lambda`, we get that the eigenvalues are:
`\lambda_1=2` and
`\lambda_2=6`, how both are positive the Hessian matrix is positive definite which means that the function
`A(d,h)` is minimum at that point.