Unformatted text preview: 1000
−x ≥ 0
1000 − x3
≥ 0 common denominator
We consider the left hand side of the inequality as our rational function r(x). We see r is
undeﬁned at x = 0, but, as in the previous example, the applied domain of the problem is
x > 0, so we are considering only the behavior of r on (0, ∞). The sole zero of r comes when
1000 − x3 = 0, which is x = 10. Choosing test values in the intervals (0, 10) and (10, ∞) gives
the following diagram.
(+) 0 (−)
0 10 We see r(x) > 0 on (0, 10), and since r(x) = 0 at x = 10, our solution is (0, 10]. In the context
of the problem, h represents the height of the box while x represents the width (and depth)
of the box. Solving h(x) ≥ x is tantamount to ﬁnding the values of x which result in a box
where the height is at least as big as the width (and, in this case, depth.) Our answer tells
us the width of the box can be at most 10 centimeters for this to happen.
3. As x → 0+ , h(x) = 1000 → ∞. This means the smaller the width x (and, in this cas...
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