Page 163 - Calculus Workbook For Dummies

147
Chapter 8: Using Differentiation to Solve Your Problems

The volume of a box equals length width height, thus
$
$
1 10 1 $ 1 $
60 = 3 height
height= .00125
This tells you that in 1 second, the height should fall .00125 feet or something very close to it.
(This process sometimes produces an exact answer and sometimes an answer with a very
small error.) Now, finally, see whether this number agrees with the answer. Your answer was
– ⁄10 inches/minute. Convert this to feet/second:
9
9
.
- ' 12 ' 60 = - .00125 It checks.
10
f . . . When it reaches a height of 60 feet, it’s moving up at a rate of 50 ft/sec. At this point,
how fast is the distance from 2nd base to the ball growing? The distance is growing 21.3
feet/second.
1. Draw your diagram and label it. See the following figure.

Hypotenuse

d
h(60)
Vertical
1st right triangle
2nd

3rd home

90 2
2. List all given rates and the rate you’re asked to figure out.
dh = 50 ft/sec
dt
dd = ?
dt
2
2 2
3. Write a formula that involves the variables: h + ` 90 2 = d
j
dh dd
4. Differentiate with respect to time: h = 2 d
2
dt dt
Like in the example, you’re missing a needed value, d. So use the Pythagorean Theorem to
get it:
2
2 2
h + ` 90 2 = d
j
2
2 2
j
60 + ` 90 2 = d
d . ! 140 .7 feet _ You can reject the negative answer.i
Now do the substitutions:
dh dd
2 h = 2 d
dt dt
dd
2 60 50 2 140 .7
$
$
= $
dt
$
$
dd = 2 60 50
dt 2 140 .7
$
. ft/sec
. 21 3