Page 88 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 71

7.63. A body weighing 8 Ib is dropped from a great height with no initial velocity. As it falls, the body encounters a force
due to air resistance proportional to its velocity. If the limiting velocity of this body is 4 ft/sec, find (a) an expres-
sion for the velocity of the body at any time t and (b) an expression for the position of the body at any time t.

7.64. A body weighing 160 Ib is dropped 2000 ft above ground with no initial velocity. As it falls, the body encounters a
force due to air resistance proportional to its velocity. If the limiting velocity of this body is 320 ft/sec, find (a) an
expression for the velocity of the body at any time t and (b) an expression for the position of the body at any time t.

7.65. A tank initially holds 10 gal of fresh water. At t = 0, a brine solution containing y Ib of salt per gallon is poured into
the tank at a rate of 2 gal/min, while the well-stirred mixture leaves the tank at the same rate. Find (a) the amount
and (b) the concentration of salt in the tank at any time t.
7.66. A tank initially holds 80 gal of a brine solution containing ^lb of salt per gallon. At t = 0, another brine solution con-
taining 1 Ib of salt per gallon is poured into the tank at the rate of 4 gal/min, while the well-stirred mixture leaves the
tank at the rate of 8 gal/min. Find the amount of salt in the tank when the tank contains exactly 40 gal of solution.

7.67. A tank contains 100 gal of brine made by dissolving 80 Ib of salt in water. Pure water runs into the tank at the rate
of 4 gal/min, and the well-stirred mixture runs out at the same rate. Find (a) the amount of salt in the tank at any
time t and (b) the time required for half the salt to leave the tank.
7.68. A tank contains 100 gal of brine made by dissolving 60 Ib of salt in water. Salt water containing 1 Ib of salt per
gallon runs in at the rate of 2 gal/min and the well-stirred mixture runs out at the same rate of 3 gal/min. Find the
amount of salt in the tank after 30 minutes.

7.69. A tank contains 40 1 of solution containing 2 g of substance per liter. Salt water containing 3 g of this substance
per liter runs in at the rate of 4 1/min and the well-stirred mixture runs out at the same rate. Find the amount of
substance in the tank after 15 minutes.
7.70. A tank contains 40 1 of a chemical solution prepared by dissolving 80 g of a soluble substance in fresh water. Fluid
containing 2 g of this substance per liter runs in at the rate of 3 I/mm and the well-stirred mixture runs out at the
same rate. Find the amount of substance in the tank after 20 minutes.

2
7.71. An RC circuit has an emf of 5 volts, a resistance of 10 ohms, a capacitance of 10~ farad, and initially a charge of
5 coulombs on the capacitor. Find (a) the transient current and (b) the steady-state current.
7.72. An RC circuit has an emf of 100 volts, a resistance of 5 ohms, a capacitance of 0.02 farad, and an initial charge on
the capacitor of 5 coulombs. Find (a) an expression for the charge on the capacitor at any time t and (b) the current
in the circuit at any time t.
7.73. An RC circuit has no applied emf, a resistance of 10 ohms, a capacitance of 0.04 farad, and an initial charge on the
capacitor of 10 coulombs. Find (a) an expression for the charge on the capacitor at any time t and (b) the current in
the circuit at any time t.

7.74. A RC circuit has an emf of 10 sin t volts, a resistance of 100 ohms, a capacitance of 0.005 farad, and no initial
charge on the capacitor. Find (a) the charge on the capacitor at any time t and (b) the steady-state current.

7.75. A RC circuit has an emf of 300 cos 2t volts, a resistance of 150 ohms, a capacitance of l/6xlO~ 2 farad, and an
initial charge on the capacitor of 5 coulombs. Find (a) the charge on the capacitor at any time t and (b) the steady-
state current.
7.76. A RL circuit has an emf of 5 volts, a resistance of 50 ohms, an inductance of 1 henry, and no initial current. Find
(a) the current in the circuit at any time t and (b) its steady-state component.
7.77. A RL circuit has no applied emf, a resistance of 50 ohms, an inductance of 2 henries, and an initial current of
10 amperes. Find (a) the current in the circuit at any time t and (b) its transient component.

7.78. A RL circuit has a resistance of 10 ohms, an inductance of 1.5 henries, an applied emf of 9 volts, and an initial current
of 6 amperes. Find (a) the current in the circuit at any time t and (b) its transient component.