Page 164 - Calculus Workbook For Dummies

148 Part III: Differentiation

5. Check whether this answer makes sense.
For this one, you’re on your own. Hint: Use the Pythagorean Theorem to calculate d ⁄50 second
1
after the critical moment. Do you see why I picked this time increment?

g . . . Five feet before the man crashes into the lamp post, he’s running at a speed of
15 miles/hour. At this point, how fast is the tip of the shadow moving? It’s moving at
25 miles/hour.
1. The diagram thing: See the following figure.

Initial Position Critical Position
Egad! Closer
Heís after and
me! Closer!!
15 ft. 15 ft.

6 ft.

b b
c c
2. List the known and unknown rates.
dc = - 15 miles/hour (This is negative because c is shrinking.) db = ?
dt dt
3. Write a formula that connects the variables.

This is another similar triangle situation, so —
height big triangle base big triangle
=
height little triangle base little triangle
15 = b
6 b - c
15 b - 15 c = 6 b
b 15
9 = c
3 = c
b 5
db dc
4. Differentiate with respect to t: 3 = 5
dt dt
5. Substitute known values.
db
5 -
3 = ^ 15h
dt
db = - 25 miles/hour
dt
Thus, the top of the shadow is moving toward the lamp post at 25 miles/hour — and is thus
gaining on the man at a rate of 10 miles/hour.
A somewhat unusual twist in this problem is that you never had to plug in the given distance of
5 ft. This is because the speed of the shadow is independent of the man’s position.
h . . . If the height of the cone-shaped pile is always equal to the radius of the cone’s base, how
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fast is the height of the pile increasing when it’s 18 feet tall? It’s increasing at 2 ⁄3 inches/min.
1. Draw your diagram: See the following figure.