A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume of the box as a function of x.
Solution:
From the diagram, we know that x is also the height of the box. Then, the length,
L=20-2xL=20−2x
and the width,
W=12-2xW=12−2x.
The volume of the box,
V=LWxV=LWx,
meaning that
V=(20-2x)(12-2x)(x)V=(20−2x)(12−2x)(x).
Expanding the brackets and simplifying leads us to,
V=4x^{3}-64x^{2}+240xV=4x 3
−64x 2
+240x
To figure out the domain, the following conditions must be true. The length,
L>0\Leftrightarrow 20-2x>0\Leftrightarrow x<10L>0⇔20−2x>0⇔x<10
and the width,
W>0\Leftrightarrow 12-2x>0\Leftrightarrow x<6, x>0W>0⇔12−2x>0⇔x<6,x>0.
Combining these leads us to, 0