Unformatted text preview: 2 − 2x) = 4x3 − 44x2 + 120x
To ﬁnd a suitable applied domain, we note that to make a box at all we need x > 0. Also the
shorter of the two dimensions of the cardboard is 10 inches, and since we are removing 2x
inches from this dimension, we also require 10 − 2x > 0 or x < 5. Hence, our applied domain
is 0 < x < 5.
2. Using a graphing calculator, we see the graph of y = V (x) has a relative maximum. For
0 < x < 5, this is also the absolute maximum. Using the ‘Maximum’ feature of the calculator,
we get x ≈ 1.81, y ≈ 96.77. The height, x ≈ 1.81 inches, the width, 10 − 2x ≈ 6.38 inches,
and the depth 12 − 2x ≈ 8.38 inches. The y -coordinate is the maximum volume, which is
approximately 96.77 cubic inches (also written in3 ).
4 this is a dangerous word...
There’s no harm in taking an extra step here and making sure this makes sense. If we chopped out a 1 inch
square from each side, then the width would be 8 inches, so chopping out x inches would leave 10 − 2x inches.
When we write V (x), it is in the context of function notation, not the volume V times the quant...
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