Download Assorted Postulates from Chapter One

A yam is put into a 200C oven. The rate of change of its temperature is directly proportional to
the difference of its temperature and the temperature of the oven.
a)
Formulate the differential equation which describes this situation.
b)
If the yam is at 20C when it is put into the oven, solve the differential equation.
c)
Find the value of the proportionality constant using the fact that after 30 minutes the
temperature of the yam is 120C.
d)
What will the temperature of the yam be after an hour in the oven?
e)
How long does it take the yam to heat up to 100C?
Dead leaves accumulate on the ground in a forest at a rate of 3 grams per square centimeter
per year. At the same time, these leaves decompose at a continuous rate of 75% a year.
Write a differential equation for the total quantity of dead leaves (per square centimeter) at time
t. Does the quantity of dead leaves increase indefinitely or does it approach an equilibrium
level? If it approaches an equilibrium level, find that level.
n
A certain chemical dissolves in water at a rate proportional to the product of the amount
undissolved and the difference between the concentration in a saturated solution and
the concentration in the actual solution. In 100 grams of a saturated solution it is known
that 50 grams of the substance are dissolved. If when 30 grams of the chemical are
agitated with 100 grams of water, 10 grams are dissolved in 2 hours, how much will be
dissolved in 5 hours?
Just formulate the appropriate differential equation and the associated initial value
problem. Don’t solve it at this time.
(Answer if you decide to solve: About 17.72 grams are dissolved after 5 hours.)
o
A tank with a capacity of 500 gallons originally contains 200 gallons of water with 100 lb.
of salt in solution. Water containing 1 lb. of salt per gallon is entering at a rate of 3
gal/min., and the mixture is allowed to flow out of the tank at a rate of 2 gal/min. Find
the amount of salt in the tank at any time prior to the instant when the solution begins to
overflow. Find the concentration (in pounds per gallon) of salt in the tank when it is on
the point of overflowing. Compare this concentration with the theoretical limiting
concentration if the tank had infinite capacity.
p
Consider a lake of constant volume V containing at time t and amount Q(t) of pollutant,
evenly distributed throughout the lake with a concentration c(t), where c(t) = Q(t)/V.
Assume that water containing a concentration k of pollutant enters the lake at a rate r,
and that water leaves the lake at the same rate. Suppose that pollutants are also added
directly to the lake at a constant rate P. Note that the given assumptions neglect a
number of factors that may, in some cases, be important; for example, the water added
or lost by precipitation, absorption, and evaporation; the stratifying effect of temperature
differences in a deep lake; the tendency of irregularities in the coastline to produce
sheltered bays; and the fact that pollutants are not deposited evenly throughout the
lake, but (usually) at isolated points around its periphery.
Formulate the differential equation that describes this (simplified) process.
10/19/2007 12:41:00 AM