Page 51 - How To Solve Word Problems In Calculus
P. 51
EXAMPLE 12
An 1800-gallon tank of water drains from the bottom in
30 minutes. According to Torricelli’s law, the volume of
water remaining in the tank after t minutes is
2
t
V = 1800 1 − 0 ≤ t ≤ 30
30
How fast is the water draining from the tank after 20 minutes?
Solution
dV t 1
= 3600 1 − −
dt 30 30
t
=−120 1 −
30
When t = 20
dV 20
=−120 1 − =−40 gallons per minute
dt 30
Water is draining at the rate of 40 gallons per minute. The
negative sign indicates that the volume of water is getting
smaller.
EXAMPLE 13
An environmental study of a small town indicates that in t
3
years the level of carbon monoxide will be q(t) = 0.005t +
2
0.02t + 0.01t + 2.5 parts per million (ppm). At what rate
will the CO level be increasing 1 and 2 years from now?
Solution
The rate of change of the CO level after t years is
2
q (t) = 0.015t + 0.04t + 0.01
After 1 year q (1) = 0.015 + 0.04 + 0.01 = 0.065 ppm per year
After 2 years q (2) = 0.06 + 0.08 + 0.01 = 0.15 ppm per year
38