Page 160 - Calculus Workbook For Dummies

144 Part III: Differentiation

b. Use the given constraint to relate x to y.
= $
Vol l w h $
x x y $
72 = $
c. Solve for y and substitute in equation from Step 2a to create a function of one variable.
72
y = 2
x
2
A x + 6 xy
=
2
A x = x + 6 x d 72 2 n
^ h
x
432
2
= x + x
3. Find the domain.
x > 0 is obvious
y > 0 is also obvious
And if you make y small enough, say the height of a proton — great box, eh? — x would have
to be astronomically big to make the volume 72 cubic inches. Technically, there is no maxi-
mum value for x.
4. Find the critical numbers.
432
2
A x = x + x
^ h
A x = 2 x - 432 x - 2
l ^ h
432
0 = 2 x - 2
x
432 2 x
2 =
x
x = 3 216
= 6
You know this number has to be a minimum because near the endpoints, say when x = .0001
or y = .0001, you get absurd boxes — either thin and tall like a mile-high toothpick or short
and flat like a square piece of cardboard as big as a city block with a microscopic lip. Both of
these would have enormous area and would be of interest only to calculus professors.
5. Finish.
x = 6, so the total area is
432 72
2
A 6 = 6 + Because y = 2
^ h
6 x
= 36 + 72 y = 2
= 108
That’s it — a 6-by-6-by-2 box made with 108 square inches of cardboard.