Problem 1: The population of a certain country has grown at a rate proportional to the number of
people in the country. At present, the country has 80 million inhabitants. Ten years ago it had 70
million. Assuming that this trend continues, find
a) an expression for the approximate number of people living in the country at any time t (taking t
= 0 to be the present time)
b) the approximate number of people who will inhabit the country at the end of the next ten-year
period.
Problem 2: A body cools from 60 °C to 50 °C in 15 minutes in air which is maintained at 30 °C. How
long will it take this body to cool from 100 °C to 80 °C in air that is maintained at 50 °C?
Problem 3: A 500-liter tank initially contains 300 liters of fluid in which there is dissolved 50 g of a
certain chemical. Fluid containing 30 g/l of the dissolved chemical flows into the tank at the rate of 4
l/min. The mixture is kept uniform by stirring, and the stirred mixture simultaneously flows out at the
rate of 2.5 l/min. How much of the chemical is in the tank at the instant it overflows?
Problem 4: Consider a cylindrical water tank of constant cross section A. Water is pumped into the
tank at a constant rate k and leaks out through a small hole of area a in the bottom of the tank. From
Torricelli’s principle in hydrodynamics it follows that the rate at which water flows through the hole
is αa√2gh, where h is the current depth of water in the tank, g is the acceleration due to gravity, and
α is a coefficient (0.5 ≤ α ≤ 1.0).
a) Show that the depth of water in the tank at any time satisfies the equation
dh/dt = (k - αa√2gh)/A
b) Determine the equilibrium depth the of water.