15 - MTH 256 Old Midterm I Problems 1. Suppose that a body...

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Unformatted text preview: MTH 256 Old Midterm I Problems 1. Suppose that a body (mass m ) is falling under the influence of gravity (gravita- tional constant = g ). Assume that the air resistance is proportional to velocity, F R =- kv . a) Derive a differential equation for the velocity v = v ( t ) of the body and solve the equation assuming that v (0) = 0. b) What is the terminal speed of the body? c) Find the height y = y ( t ) of the body at time t provided that y (0) = H . 2. A tank with capacity of 500 liters originally contains 200 liters of water with 10 kg of salt in solution. Water containing 0.1 kg of salt per liter is entering at a rate of 3 liters per minute and the well-stirred mixture is allowed to flow out of the tank at a rate of 2 liters per minute. a) Derive a differential equation for the amount of salt Q = Q ( t ) in the tank. b) Solve the differential equation to find an expression for the amount of salt in the tank at time t . c) What is the amount of salt in the tank at the moment when it starts over- flowing? 3. A circular cone (height H = 2 m , radius of the top R = 1 m ) with a vertical axis and the vertex pointing down is filled with water. At t = 0 a small circular hole is opened at the vertex. By Torricellis law, the rate of change of the volume V = V ( t ) of water in the cone is (1) dV dt =- 2 y, where y = y ( t ) is the depth of water. a) Express the volume of water V ( t ) in terms of the depth of water y ( t ). (Draw a picture!) b) Use equation (1) to find an equation for the rate of change of y ( t ) and solve this equation. c) How long does it take for all water to drain from the tank? 4. A cone (height = 2ft) is filled with water. At t = 0 a small circular hole is opened at the vertex. By Torricellis law, the rate of change of the volume V = V ( t ) of water in the cone is (1) dV dt =- 2 y, where y = y ( t ) is the depth of water.) is the depth of water....
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This note was uploaded on 11/05/2009 for the course ECON 124 taught by Professor Washingtonm during the Spring '09 term at Foothill College.

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15 - MTH 256 Old Midterm I Problems 1. Suppose that a body...

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