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A tank initially holds 100 gallons of brine solution containing 20 lbs. of salt. At t = 0, fresh water is poured into the tank at the rate of 5 gal/min., while the well- stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time t.

A tank initially holds 100 gallons of brine solution containing 20 lbs. of salt. At t = 0, fresh water is poured into the tank at the rate of 5 gal/min., while the well- stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time t.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. A tank initially holds 100 gallons of brine solution containing 20 lbs. of salt. At t = 0, fresh water is poured into the tank at the rate of 5 gal/min., while the well- stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time t. 2. A tank 100 gallon capacity is initially full of water. Pure water is allowed to run into the tank at a rate of 1 gallon per minute, and at the same time containing one-fourth pound of salt per gallon flows into the tank also at a rate of 1 gallon per minute. The mixture flows out at a rate of 2 gallons per minute (it is assumed that there is perfect mixing). Find the amount of salt in the tank after t minutes. 3. The rate of growth of an investment is proportional to the amount of the investment at any time t. The initial investment is $1000 and after 10 years the balance is $3320.12. What is the particular solution? 4. The limiting capacity of the habitat of wildlife herd is 750. The growth rate dN/dt of the herd is proportional to the unutilized opportunity for the growth. As describe by differential equation dN/dt = k(750 – N). When t = 0, the population of the herd is 100. After two years the population has growth to 160. Write the population N as a function of t. What is the population of the herd after 4 years? %3D 5. An apple pie with an initial temperature of 170 °C is removed from the oven and left to cool in a room temperature of 20 °C. Given that the temperature of the pie initially decreases at a rate of 3.0 °C/min., how long will it take for the pie to cool to a temperature of 30 °C? 6. A circuit consists of an inductance of 0.05 henry, a resistance of 5 ohms, and a condenser of capacitance 4 x 10-4 farad. If q = i = 0 when t = 0, find q and i in terms of t when (a) there is a constant emf = 110 volts, (b) there is an alternating emf = 200 cos100t.
Transcribed Image Text:1. A tank initially holds 100 gallons of brine solution containing 20 lbs. of salt. At t = 0, fresh water is poured into the tank at the rate of 5 gal/min., while the well- stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time t. 2. A tank 100 gallon capacity is initially full of water. Pure water is allowed to run into the tank at a rate of 1 gallon per minute, and at the same time containing one-fourth pound of salt per gallon flows into the tank also at a rate of 1 gallon per minute. The mixture flows out at a rate of 2 gallons per minute (it is assumed that there is perfect mixing). Find the amount of salt in the tank after t minutes. 3. The rate of growth of an investment is proportional to the amount of the investment at any time t. The initial investment is $1000 and after 10 years the balance is $3320.12. What is the particular solution? 4. The limiting capacity of the habitat of wildlife herd is 750. The growth rate dN/dt of the herd is proportional to the unutilized opportunity for the growth. As describe by differential equation dN/dt = k(750 – N). When t = 0, the population of the herd is 100. After two years the population has growth to 160. Write the population N as a function of t. What is the population of the herd after 4 years? %3D 5. An apple pie with an initial temperature of 170 °C is removed from the oven and left to cool in a room temperature of 20 °C. Given that the temperature of the pie initially decreases at a rate of 3.0 °C/min., how long will it take for the pie to cool to a temperature of 30 °C? 6. A circuit consists of an inductance of 0.05 henry, a resistance of 5 ohms, and a condenser of capacitance 4 x 10-4 farad. If q = i = 0 when t = 0, find q and i in terms of t when (a) there is a constant emf = 110 volts, (b) there is an alternating emf = 200 cos100t.
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