Examples--Application Problems

Example 15:  Exponential Population Growth Example

Exponential Population Growth The population of a country is growing at a rate that is proportional to the population of the country.  The population in 1990 was 20 million and in 2000 the population was 22 million.  Estimate the population in 2020.

Solution

Example 16:  Solution to a mixture problem:  We start with a tank containing 50 gallons of salt water with the salt concentration being 2 lb/gal.  Salt water with a salt concentration of 3 lb/gal is then poured into the top of the tank at the rate of 3 gal/min and salt water is at the same time drained from the bottom of the tank at the rate of 3 gal/min.  We will consider the water and salt mixture in the tank to be well-stirred and at all times to have a uniform concentration of salt.  Find the function S(t) that gives the amount of salt in the tank as a function of time (t) since we began pouring in salt water at the top and simultaneously draining salt water from the bottom of the tank.  How long before there will be 120 pounds of salt in the tank?

Graph of the solution function

Example 17:  Solution to a Newton's Law of Cooling problem:  A pot of liquid is put on the stove to boil.  The temperature of the liquid reaches 170^oF and then the pot is taken off the burner and placed on a counter in the kitchen.  The temperature of the air in the kitchen is 76^oF.  After two minutes the temperature of the liquid in the pot is 123^oF.  How long before the temperature of the liquid in the pot will be 84^oF?  Click here for more on Newton's Law of Cooling.

Solution Function Graph

Example 18:  SAILING:   Ignoring resistance, a sailboat starting from rest accelerates (dv/dt) at a rate proportional to the difference between the velocities of the wind and the boat.  (a)  Write the velocity as a function of time if the wind is blowing at 20 ft/sec and after one second the boat is moving at 5 ft/sec.  Assume the boat started from rest.  (b)  Use the result in part (a) to write the distance traveled by the boat as a function of time.  Differential Equations Sailing Application Solution  (Equation similar to that governing Newton's Law of Cooling)

Example 19:  An Application Involving a Snowplow

On a Tuesday morning in February before noon in rural Minnesota it started to snow.  There had been no snow on the ground before it started to snow.  Literally immediately it was snowing at a steady, constant rate so that the thickness of the snow on the ground was increasing at a constant rate.  A snowplow began clearing the snow from the streets at noon.  The speed of the snowplow in clearing the snow is inversely proportional to the thickness of the snow.  The snowplow traveled two miles during the first hour after noon and traveled one mile during the second hour after noon.  At what time did it begin snowing?

 

a = The length of time before noon when it started snowing. t = The length of time the snowplow had been traveling (and plowing). v = The speed of the snowplow. s = The distance traveled by the snowplow. T = The thickness of the snow.

Click on the picture to see a snowplow in action (with sound).  Click here for more action (takes longer to load).

s(0) = 0,     s(1) = 2,     s(2) = 3

v = k/(t + a)    since the speed of the snowplow in clearing the snow is inversely proportional to the thickness of the snow.

Thus

Here is an animation showing the snowplow plowing (and slowing down).  Quicktime version

Solution

MIXTURE PROBLEM 2:  A tank holds 100 gallons of liquid.  The tank is half full with a salt water solution that contains 0.1 lb of salt per gallon.  Pure water is added to the container at the rate of 2 gallons per minute and at the same time one gallon of salt water per minute is removed from the tank.  Assume that the concentration of salt in the water in the tank remains uniform throughout.  When the tank becomes full it begins to overflow and at that time a total of 2 gallons per minute of salt water will be leaving the tank.  Construct a piecewise continuous function S(t) that gives the amount of salt in the tank as a function of time t where t = 0 represents the time when the 2 gallons per minute of pure water began being added to the tank and the 1 gallon per minute of "well-stirred, uniform" mixture began exiting the tank.  The picture at the right shows the amount of salt in the tank as a function of time.  Solution

NEWTON'S LAW OF COOLING PROBLEM 2:  When a thermometer reads 36oF, it is placed in an oven.  After 1 and 2 minutes, respectively, it reads 60oF and 82oF.  What is the temperature of the oven?  Justify your answer analytically.  Solution

NEWTON'S LAW OF COOLING PROBLEM 3:  Some hot chocolate has been created using milk and chocolate and has a temperature of 200oF.  The hot chocolate is in a cup that is 9/10 full.  Cool milk at a temperature of 50oF is to be used to help cool the hot chocolate.  Assume that the cooling constant, k, from Newton's Law of Cooling is the same for the hot chocolate and the milk and therefore the same for any mixture of hot chocolate and milk.  If the cup of 200oF hot chocolate is filled the rest of the way up to the brim with the 50o milk then the milk diluted hot chocolate will have an immediate temperature of Two minutes later the milk diluted hot chocolate will have a temperature of 160oF if it is in a room whose temperature is 80oF.  Use this fact to calculate k.  (a)  If the milk was added to the hot chocolate at time t=0 minutes, how long will it take for the temperature of the milk diluted hot chocolate to reach 120oF?  (b)  Suppose instead that the milk is not added to the hot chocolate until the originally 200oF hot chocolate has been in the 80oF room for 5 minutes.  In this case how long will it take (total) for the hot chocolate (ultimately diluted by milk) to reach a temperature of 120oF?  (c)  Using the 80oF room and the 50oF milk, how could the hot chocolate be cooled to 120oF in the least amount of time and what would this time be?  Solution

Red gives the temperature curve when the milk is added immediately. Blue gives the temperature curve when the milk is not added until five minutes after the hot chocolate was poured into the cup in the 80oF room. A similar problem can be found at the Joseph Mahaffy site linked to in the 3.1 and 3.2 notes under Lectures--Linear Differential Equations--Worked Examples.

HEAT SEEKING PARTICLE EXAMPLES:  Click here to see the heat seeking particle examples from Calculus III.  You will also find Maple worksheets describing how to find the path of steepest descent down a mountain.

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats