Page 159 - Calculus Workbook For Dummies

143
Chapter 8: Using Differentiation to Solve Your Problems

c. Solve for one variable in terms of the other — take your pick — and substitute into your
formula to create a function of a single variable.
1
2 h = ^ 5h A = 2 bh
b h -
2 h = bh - 5 b 1 5 b
A b = b $ c m
^ h
b = -
h 2 - h 5 b 2 b - 2
^
5 b 2
5 b =
h = 2 b - 4
b - 2
3. Find the domain.
b must be greater than 2 — do you see why? And there’s no maximum value for b.
4. Find the critical numbers.
5 b 2
A b =
^ h
2 b - 4
2 l
4 l
2
2
4 - _
_ 5 b i 2 ^ b - h 5 b ^ i 2 b - h 10 b - 40 b
A b = 2 2 = 0
l ^ h
2 ^ b - 4h 2 ^ b - 4h
2
b 0
10 ^ b 4 - 10 b 2 10 b - 40 =
b 2 - h
= 2
b b - h
2 ^ b - 4h 10 ^ 4 = 0
2
=
10 b - 40 b b 0 or 4
= 2
2 ^ b - 4h
Zero is outside the domain, so 4 is the only critical number. The smallest triangle must occur
at b = 4 because near the endpoints you get triangles with astronomical areas.
5. Finish.
b 4
=
5 b
h = so
b - 2
5 4 $
h = = 10 ;
4 - 2
And the triangle ’s area is thus 20 .
d . . . Given that you want a box with a volume of 72 cubic inches, what dimensions will mini-
mize the total cardboard area and thus minimize the cost of the cardboard? The minimizing
dimensions are 6-by-6-by-2, made with 108 square inches of cardboard.
1. Draw a diagram and label with variables (see the following figure).
y

Mixed Nuts x
For Dummies
x

2. a. Express the thing you want to minimize, the cardboard area, as a function of the variables.
square base four sides two dividers
H F H
Cardboard area= x 2 + 4 xy + 2 xy
2
A x + 6 xy
=