11DIFFERENTIATION - Department of Mathematics

SOLUTION ✔
11.2 THE PRODUCT AND QUOTIENT RULES 713
a. Derive a general expression that gives the rate of change of the pond’s
oxygen level at any time t.
b. How fast is the pond’s oxygen content changing 1 day, 10 days, and 20
days after the organic waste has been dumped?
a. The rate of change of the pond’s oxygen level at any time t is given by the
derivative of the function f. Thus, the required expression is
f(t) 100 d
dtt2 10t 100
t2 20t 100
100 (t 2 20t 100) d
dt (t 2 10t 100) (t 2 10t 100) d
dt (t 2 20t 100)
(t 2 20t 100) 2
100 (t 2 20t 100)(2t 10) (t 2 10t 100)(2t 20)
(t 2 20t 100) 2
1002t 3 10t 2 40t 2 200t 200t 1000 2t 3 20t 2 20t 2 200t 200t 2000
(t 2 20t 100) 2
100 10t 2 1000
(t 2 20t 100) 2
b. The rate at which the pond’s oxygen content is changing 1 day after the
organic waste has been dumped is given by
10 1000
f(1) 100(1 20 100) 2 6.76
That is, it is dropping at the rate of 6.8% per day. After 10 days the rate is
f(10) 100 10(10)2 1000
(100 200 100) 2 0
That is, it is neither increasing nor decreasing. After 20 days the rate is
f(20) 100 10(20)2 1000
(400 400 100) 2 0.37
That is, the oxygen content is increasing at the rate of 0.37% per day, and the
restoration process has indeed begun.