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Unformatted text preview: → ∞. Readjusting the window, we ﬁnd S does possess a relative minimum
at x ≈ 12.60. As far as we can tell,4 this is the only relative extremum, and so it is the
absolute minimum as well. This means the width and depth of the box should each measure
approximately 12.60 centimeters. To determine the height, we ﬁnd h(12.60) ≈ 6.30, so the
height of the box should be approximately 6.30 centimeters. 4.3.1 Variation In many instances in the sciences, rational functions are encountered as a result of fundamental
natural laws which are typically a result of assuming certain basic relationships between variables.
These basic relationships are summarized in the deﬁnition below.
Definition 4.4. Suppose x, y , and z are variable quantities. We say
• y varies directly with (or is directly proportional to) x if there is a constant k such
that y = kx.
• y varies inversely with (or is inversely proportional to) x if there is a constant k such
that y = x .
• z varies jointly with (or is jointly proportional to) x and y if there is a cons...
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