Question

 You plan to make a simple, open topped box from a piece of sheet metal by cutting a square – of equal size – from each corner and folding up the sides as shown in the diagram: If 𝑙 = 200𝑚𝑚 and 𝑤 = 150𝑚𝑚 calculate: a) The value of x which will give the maximum volume b) The maximum volume of the box c) Comment of the value obtained in part b.

Answer 1

Solution

a) The value of x which will give the maximum volume

Given that L:= 200 ; W:= 150

The volume of the box is $V=lwh$, where length $l=L-2x$, width $w=W-2x$, and height h=x. Therefore the volume of the box can be written in the form: $V(x)=(L - 2\cdot x)\cdot(W- 2\cdot x)\cdot x$

Lengths and width of the box decreased that is of sheet metal by $x$ from each corner, and height of the box is equal $x$. We bring $V(x)$ to a simple form:

$$V(x)=4\cdot x^3 -2\cdot (L+W)\cdot x^2+ L\cdot W\cdot x$$

To find maximum volume one compute the derivative of volume with respect to $x$

$V^{'}_x=12\cdot x^2 - 4\cdot (L+W)\cdot x+ L\cdot W$ and define the root of the equation $V^{'}_x=0$ :

$x_{1,2}=(2\cdot(L+W)\pm\sqrt{4(L+W)^2-12\cdot L\cdot W} )/12=\frac{1}{6}(L+W\pm\sqrt{L^2+W^2-L\cdot W})$

$x_1=88.38 ;\space x_2=28.29$

The first value cannot be implemented. It is clear that the box will succeed only if $x<W/2$ . The second value corresponds to the maximum volume shown in the image below.

Answer:

$$x=28.29 mm; max V=379037.81 mm^2$$