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Unformatted text preview: = 80, we get 150 = 0, which is impossible, so we conclude
that AC (x) > 80 for all x > 0. This means the average cost per system is always greater
than $80 per system, but the average cost is approaching this amount as more and more
systems are produced. Looking back at Example 2.1.5, we realize $80 is the variable cost
per system − the cost per system above and beyond the ﬁxed initial cost of $150. Another
way to interpret our answer is that ‘inﬁnitely’ many systems would need to be produced to
eﬀectively counterbalance the ﬁxed cost.
Our next example is another classic ‘box with no top’ problem.
Example 4.3.3. A box with a square base and no top is to be constructed so that it has a volume
of 1000 cubic centimeters. Let x denote the width of the box, in centimeters. Refer to the ﬁgure
below. height depth
width, x 1. Express the height h in centimeters as a function of the width x and state the applied domain.
2. Solve h(x) ≥ x and interpret.
3. Find and interpret the behavior of h(x) as x → 0+ and as x → ∞.
4. Express the surface area S of the box as a...
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