Page 156 - Calculus Workbook For Dummies

140 Part III: Differentiation

Solutions to Differentiation Problem Solving

a What are the dimensions of the soup can of greatest volume that can be made with 50 square
inches of tin? What’s its volume? The dimensions are 3 ⁄4 inches wide and 3 ⁄4 inches tall. The
1
1
volume is 27.14 cubic inches.
1. Draw your diagram (see the following figure).
r

h

Soup For
Dummies

2
2. a. Write a formula for the thing you want to maximize, the volume: V π= r h
b. Use the given information to relate r and h.
top and bottom lateral area
4
6 7 8 H
4
4
4
Surface Area= 2 r π 2 + 2 π rh
2
50 2 r π + 2 π rh
=
2
25 = r π + π rh
c. Solve for h and substitute to create a function of one variable.
π rh 25 - r π 2 V π r h
2
=
=
25 25
2
h = r π - r V r = r π c r π - r m
^ h
= 25 r - r π 3
3. Figure the domain.
r > 0 is obvious
h > 0 is also obvious
2
And because 25 = r π + π rh (from Step 2b), when h = 0, r = 25 so r must be less than 25 , or
π
π
about 2.82 inches.
4. Find the critical numbers of V r ^ h.
V r = 25 r - r π 3
^ h
r =
V l ^ h 25 - 3 r π 2
0 = 25 - 3 r π 2
25
2
r =
3 π
25
r = !
3 π
. . 1 63 inches You can reject the negative answer because ’ its outside the domain .i
_
5. Evaluate the volume at the critical number.
π
V ^ . 1 63 = 25 $ . 1 63 - ^ . 1 63h 3
h
. 27 .14 cubic inches