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A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 18 in. by 30 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 V of the box as a function of x. : If squares with a side x, are removed from each corner, the dimensions of the box: (30-2x) by (18-2x) by x : V = H * L * W : V = x*(18-2x)*(30-2x) FOIL V = x(540 - 36x - 60x + 4x^2) : V = x(4x^2 - 96x + 540) :
V(x) = 4x^3 - 96x^2 + 540x
Solution:
From the diagram, we know that x is also the height of the box. Then, the length,
L=20-2xand the width,
W=12-2x.
The volume of the box,
V=LWx,
meaning that
V=(20-2x)(12-2x)(x).
Expanding the brackets and simplifying leads us to,
V=4x^{3}-64x^{2}+240xTo figure out the domain, the following conditions must be true. The length,
L>0\Leftrightarrow 20-2x>0\Leftrightarrow x<10and the width,
W>0\Leftrightarrow 12-2x>0\Leftrightarrow x<6, x>0.
Combining these leads us to, 0<x<6
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