Mapua Institute of TechnologyDepartment of Mathematics

MATH24-1 Project(Solved Problems on Application of First Order, First Degree Differential Equation)

Submitted by: Alejo, Christian Allen S. Math 24 - 1/ A3Law of Growth and Decay

1. Almost all the radon in the world today was created within the past week or so by a chain of radioactive decays beginning mainly from uranium, which has been part of the earth since it was formed. This cascade of decaying elements is quite common, and in this problem we study a toy model in which the numbers work out decently. This is about Tatooine, a small world endowed with unusual elements. A certain isotope of Startium, symbol St, decays with a half-life ts. Strangely enough, it decays with equal probability into a certain isotope of either Midium, Mi, or into the little known stable element Endium. Midium is also radioactive, and decays with half-life tm into Endium. All the St was in the star-stuff that condensed into Tatooine, and all the Mi and En arise from the decay route described. Also, tm = ts. Use the notation x(t), y(t), and z(t), for the amount of St, Mi, and En on Tatooine, in units so that x(0) = 1.Write down the differential equations controlling x, y, and z. Be sure to express the constants that occur in these equations correctly in terms of the relevant decay constants. Use the notation (Greek letter sigma) for the decay constant for St and (Greek letter mu) for the decay constant for Mi. Your first step is to relate to ts and to tm. A check on your answers: the sum x+y+z is constant, and so we should have x+y+z = 0. Solve these equations, successively, for x, y, and z.

2. A tree contains a known percentage p0 of a radioactive substance with half-life t. When the tree dies the substance decays and isnt replaced. If the percentage of the substance in the fossilized remains of such a tree is found to be p1, how long has the tree been dead?

3. Suppose a substance decays at a yearly rate equal to half the square of the mass of the substance present. If we start with 50 g of the substance, how long will it be until only 25 g remain?

Newtons Second Law of Motion

1. An object of mass m is released from rest and falls under the influence of gravity. If the magnitude of the force due to air resistance is bvn, where b and n are positive constants, find the limiting velocity of the object (assuming this limit exists). [Hint: Argue that the existence of a (finite) limiting velocity implies that dv/dt 0 as t infinity .]

2. A model rocket having initial mass m0 kg is launched vertically from the ground. The rocket expels gas at a constant rate of kg/sec and at a constant velocity of m/sec relative to the rocket. Assume that the magnitude of the gravitational force is proportional to the mass with proportionality constant g. Because the mass is not constant, Newtons second law leads to the equation

where v = dx/dt is the velocity of the rocket, x is its height above the ground, and mo - t is the mass of the rocket at t sec after launch. If the initial velocity is zero, solve the above equation to determine the velocity of the rocket and its height above ground for 0m0/

3. An object weighing 256 lb is dropped from rest in a medium that exerts a resistive force with magnitude proportional to the square of the speed. The magnitude of the resisting force is 1 lb when |v| = 4 ft/s. Find v for t > 0, and find its terminal velocity.

Mixture problem

1. A right circular cone is filled with water. What time will the water empty through an orifice O of cross-sectional area a at the vertex? Assume velocity of exit is v=k where h is the instantaneous height of the water level above O and k is the discharge coefficient.

2. A tank holds V liters of salt water. Suppose that a saline solution with concentration of c grams/liter is added at the rate of r liters/minute. A mixer keeps the salt essentially uniformly distributed in the tank. A pipe lets solution out of the tank at the same rate of r liters/minute. Write down the differential equation for the amount salt in the tank.

Use the notation x(t) for the number of grams of salt in the tank at time t. Check the units in your equation. Write it in standard linear form.

3. Assume that c and r are constant in the previous problem. In fact, suppose that r = 2 liters/minute and the volume of the tank is V = 1 liter. Solve this equation under the assumption that x(0)= 0. What is the limiting amount of salt in the tank? Does your result make sense by simple logic? When will the tank contain half that amount? Simple interest1. A benefactor wishes to establish a trust fund to pay a researchers salary for T years. The salary is to start at S0 dollars per year and increase at a fractional rate of a per year. Find the amount of money P0 that the benefactor must deposit in a trust fund paying interest at a rate r per year. Assume that the researchers salary is paid continuously, the interest is compounded continuously, and the salary increases are granted continuously. 2. A bank pays interest continuously at the rate of 6%. How long does it take for a deposit of Q0 to grow in value to 2Q0?

3. A person deposits $25,000 in a bank that pays 5% per year interest, compounded continuously. The person continuously withdraws from the account at the rate of $750 per year. Find V(t), the value of the account at time t after the initial deposit.

Logistic growth1. For the logistic curve (15), assume and pa := p(ta) and pb := p(tb) are given with tb =2ta (ta>0).Show that:

2. The state game commission releases 100 deer into a game preserve. During the first 5 years the population increases to 450 deer. Find a model for the population growth assuming logistic growth with a limit of 5000 deer. What does the model predict the size of the population will be in 10 years, 20 years, 30 years?

3. Glucose is being fed intravenously to the bloodstream of a patient at 0.01 grams per minute. At the same time, the patients body converts the glucose and removes it from the bloodstream at a rate proportional to the amount of glucose present. a. Let g(t) be the amount of glucose in the bloodstream at time t in minutes. Let k be the proportionality constant mentioned above. Set up a differential equation for g(t). b. Find the general solution to this differential equation. Show that the amount of glucose in the bloodstream always approaches (0.01/k) as t becomes very large. c. Suppose that there are 4.1grams of glucose in the bloodstream at t=0 and that as t becomes very large, the glucose level approaches 5.2grams. How much glucose is in the blood one hour after starting?

Newtons Law of Cooling

1. Early Monday morning, the temperature in the lecture hall has fallen to 40F, the same as the temperature outside. At 7:00 A.M., the janitor turns on the furnace with the thermostat set at 70F. The time constant for the building is 1/k = 2 hr and that for the building along with its heating system is 1/k1 = 1/2 hr. Assuming that the outside temperature remains constant, what will be the temperature inside the lecture hall at 8:00 A.M.? When will the temperature inside the hall reach 65F?

2. Two friends sit down to talk and enjoy a cup of coffee. When the coffee is served, the impatient friend immediately adds a teaspoon of cream to his coffee. The relaxed friend waits 5 min before adding a teaspoon of cream (which has been kept at a constant temperature). The two now begin to drink their coffee. Who has the hotter coffee? Assume that the cream is cooler than the air and use Newtons law of cooling.

3. On a mild Saturday morning while people are working inside, the furnace keeps the temperature inside the building at 21C. At noon the furnace is turned off, and the people go home. The temperature outside is a constant 12C for the rest of the afternoon. If the time constant for the building is 3 hr, when will the temperature inside the building reach 16C? If some windows are left open and the time constant drops to 2 hr, when will the temperature inside reach 16C?

Escape Velocity

1. The radius of the moon is roughly 1080 miles. The acceleration of gravity at the surface of the moon is about 0.165g, where g is the acceleration of gravity at the surface of the earth. Determine the velocity of escape for the moon. (Hint: g = 32.16 ft/sec2and 1 mile = 5280 ft)

2. The sun is 330,000 times more massive and has a radius 109 times that of the earth. How small would the Suns radius have to shrink for it to become a black hole: that is, to where its escape velocity is 3 x 10 8 m/s ?

3. A space vehicle is to be launched from the moon, which has a radius of about 1080 miles. The acceleration due to gravity at the surface of the moon is about 5:31 ft/s2. Find the escape velocity in miles/s.

References: http://ocw.mit.edu (problem 1, law of growth and decay; problems 2 and 3, mixture problem) Elementary Differential Equations by William Trench (problems 2 and 3, law of growth and decay; problem 3, Newtons second law of motion; problems 1, 2 ,3 ,simple interest) Fundamentals of Differential Equations by Nagle e.t., al (problems 1 and 2 - Newtons second law of motion; problems 2 and 3 - mixture problem; problems 1,2 and 3 - logistic growth; problems 1, 2 and 3 - Newtons law of cooling) http://www.math-principles.com (problems 1 and 2 escape velocity)