Page 106 - Calculus Demystified

CHAPTER 3 Applications of the Derivative
At the instant the problem is posed, b(t) = 5, h(t) = 12 (by the Pythagorean 93
theorem), and b (t) = 1. Substituting these values into the equation yields

5 · 1 5

h (t) =− =− ft/min.
12 12
Observe that the answer is negative, which is appropriate since the top of the
ladder is falling.

You Try It: Suppose that a square sheet of aluminum is placed in the hot sun.
It begins to expand very slowly so that its diagonal is increasing at the rate of
1 millimeter per minute. At the moment that the diagonal is 100 millimeters, at
what rate is the area increasing?
EXAMPLE 3.13
A sponge is in the shape of a right circular cone (Fig. 3.16). As it soaks up
water, it growsin size. At a certain moment, the height equals6 inches, and
is increasing at the rate of 0.3 inches per second. At that same moment, the
radiusis4 inches, and isincreasing at the rate of 0.2 inchesper second.
How isthe volume changing at that time?

Fig. 3.16

SOLUTION
We know that the volume V of a right circular cone is related to the height
h and the radius r by the formula
1
2
V = πr h.
3
Differentiating both sides with respect to the variable t (for time in seconds)
yields

dV 1 dr 2 dh
= π 2r h + r .
dt 3 dt dt