Page 158 - Calculus Workbook For Dummies

142 Part III: Differentiation

4. Find the critical numbers of C xh.
^
C x = 15 π x + 40 x + 400
^ h
x
x =
^
Cl ^ h 15 π + 40 + - 400h x - 2
0 = 15 π + 40 - 400 x - 2
2
-
400 x = 15 π + 40
400
2
x =
15 π + 40
400
x = !
15 π + 40
x . ! . 2 143
Omit –2.143 because it’s outside the domain. So 2.143 is the only critical number.
5. Evaluate the cost at the critical number and at the endpoints.
C x = 15 π x + 40 x + 400
^ h
x
C 0 = undefined
^ h
h
C ^ . 2 143 . $373
C ^ . 3 57 . $423
h
So, the least expensive frame for a 20-square-foot window will cost about $373 and will be
10 π x
2 × 2.143, or about 4.286 feet or 4’3’’ wide at the base. Because y = - , the height of the
x 4
rectangular lower part of the window will be 2.98, or about 3’ tall. The total height will thus
be 2.98 plus 2.14, or about 5’1’’.
c . . . Given that a right triangle’s hypotenuse must pass through the point (2, 5), what are the
dimensions and area of the smallest such triangle? The hypotenuse meets the y-axis at (0, 10)
and the x-axis at (4, 0) and the triangle’s area is 20.
1. Draw a diagram (see the following figure).

y

5
(2,5)
4
h
3
2
1
x
1 2 3 4 5
b
1
2. a. Write a formula for the thing you want to minimize, the area: A = bh
2
b. Use the given constraints to relate b and h.
This is a bit tricky — Hint: consider similar triangles. If you draw a horizontal line from (0, 5)
to (2, 5), you create a little triangle in the upper-left corner that’s similar to the whole triangle.
(You can prove their similarity with AA — remember your geometry? — both triangles have a
right angle and both share the top angle.)
Because the triangles are similar, their sides are proportional:
height big triangle height small triangle
=
base big triangle base small triangle
h = h - 5
b 2