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Unformatted text preview: e, depth),
the larger the height h has to be in order to maintain a volume of 1000 cubic centimeters. As
x → ∞, we ﬁnd h(x) → 0+ , which means to maintain a volume of 1000 cubic centimeters,
the width and depth must get bigger the smaller the height becomes.
3 That is, h(x) means ‘h of x’, not ‘h times x’ here. 272 Rational Functions 4. Since the box has no top, the surface area can be found by adding the area of each of the
sides to the area of the base. The base is a square of dimensions x by x, and each side has
dimensions x by h. We get the surface area, S = x2 + 4xh. To get S as a function of x, we
substitute h = 1000 to obtain S = x2 + 4x 1x2 . Hence, as a function of x, S (x) = x2 + 4000 .
The domain of S is the same as h, namely (0, ∞), for the same reasons as above.
5. A ﬁrst attempt at the graph of y = S (x) on the calculator may lead to frustration. Chances
are good that the ﬁrst window chosen to view the graph will suggest y = S (x) has the x-axis
as a horizontal asymptote. From the formula S (x) = x2 + 4000 , however, we get S (x) ≈ x2 as
x → ∞, so S (x)...
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