Question

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

Since x is the side of the square that has to be cutted from each corner in order to make a box, then the volume of the box can be expressed in terms of x following way:

$V(x) = (l-2x)*(w-2x)*x = (200-2x)*(150-2x)*x$

The point is to find such value of x that maximizing V(x). We should find derivative of the V(x)

$V'(x) = 3000-800x-600x+12x^{2}=12x^{2}-1400x+30000$ and find such x that V'(x) = 0

$12x^{2}-1400x+30000=0\to 3x^{2} - 350x+7500=0$

After solving this equation we received $x{\scriptscriptstyle 1}=28.29mm, x{\scriptscriptstyle 2}=88.38mm$. Second root doesn't satisfy the conditions of the task, cause x cannot be greater than ${\frac w 2} = 75$. In point $x{\scriptscriptstyle 1}$ V'(x) change sign from + to -, so it's a point of maximum of V(x).

$V(x{\scriptscriptstyle 1})=379037.81mm^{2}$

So, we find out that if you have a rectangular with sizes 150mmX200mm(or 0.15mX0.2 m) you can make a box with the max roomines of 379037.81$mm^{2}$ (or approximately 0.379$m^{2}$)