7.2 - Exponential Growth and Decay

Transcription

1 Lecture_07_02.nb Exponential Growth and Decay Introduction Population growth, compound interest, radioactive decay, and heating and cooling can be described by differential equations. The Law of Exponential Change Suppose that a quantity grows or decays at a rate that is proportional to the amount present and that the initial amount is yh0l = y 0. Then the differential equation that models this situation is dy ÅÅÅÅÅ dt = y, yh0l = y 0 To solve this differential equation, first note that it is separable. So we can write Now we can integrate both sides. dy ÅÅÅÅÅ y = dt Solve for y. Ÿ 1 ÅÅÅÅ y y = Ÿ t ln» y» ã t + C» y» ã t+ C = t C = C 1 t where C 1 = C Recall that» y» ã y if y 0 and» y» =-y if y < 0. Thus we have or y ã C 1 t where C 1 = C y ã C 1 t where C 1 = - C In either case, C 1 will be determined by the initial condition. Recall that yh0l = y 0. y 0 = yh0l ã C 1 0 = C 1 0 = C 1 H1L = C 1

2 Lecture_07_02.nb 2 Thus y = y 0 t If > 0 then we have exponential growth, and if < 0 we have exponential decay. The Law of Exponential Change y = y 0 t Growth: > 0 Decay: < 0 The number is the rate constant of the equation. We can use this model to describe population growth or radioactive decay. Unlimited Population Growth According to United Nations data, the world population in 1998 was approximately 5.9 billion and growing at a rate of about 1.33% per year. Assuming an exponential growth model, estimate the world population at the beginning of the year 2004.

3 Lecture_07_02.nb 3 Short term, the exponential growth model does a good job modeling the population growth. Long term, the model is inadequate. The graph of the growth function is given by For this model, growth is unbounded, which is unrealistic. Real life places limits on the amount of resources available (food, clothing, shelter, water, and land). At some point the environment cannot sustain the population any more. We will discuss a better model in Section 9.5. Continuously Compounded Interest If you invest P dollars at R percent (where r is the rate as a decimal) compounded times per year for t years, then the future value (interest and principle) is given by A = PH1 + r ÅÅÅÅ Lt We can thin of A as a function of time and the principle as the initial condition, that is A 0 = P. Thus, we can write A HtL = A 0 H1 + r ÅÅÅÅ Lt If $1000 is deposited at 10%, how much will you have after 1 year if interest is compounded: (a) annually, (b) quarterly, (c) monthly, (d) daily, (e) hourly, (f) every minute, and (g) every second.

4 Lecture_07_02.nb 4 AH1L This function has a limit as Ø. lim Ø AHtL = lim Ø A 0 H1 + r ÅÅÅÅ Lt = A 0 lim Ø H1 + r ÅÅÅÅ = A 0 lim Ø A H1 + r ÅÅÅÅ As Ø, ÅÅÅÅ r Ø 0. Let x = ÅÅÅÅ r, then ÅÅÅÅ 1 L ÅÅÅÅ r ÿrt L ÅÅÅÅ r E rt = ÅÅÅÅ x r ÅÅÅÅ lim Ø AHtL = A 0 lim xø0 AH1 + xl 1 rt x E = A 0 rt and we have Thus, if we deposit A 0 dollars into an account that pays r percent compounded continuously for t years, the future value is AHtL = A 0 rt If $2500 is deposited at 8%, how much will you have after 5 years if interest is compounded continuously.

5 Lecture_07_02.nb 5 Radioactivity Atoms can emit radiation through a process of radioactive decay. In some cases, the atom will cahnge from one element to another. We can use the law of exponential change to model radioacitve decay. Radon-222 is a radioactive gas with a half-life of 3.83 days. This gas is a health hazard because it tends to get trapped in the basements of houses, and many health officials suggest that homeowners seal their basements to prevent entry of the gas. Assume that 5.0 µ 10 7 radon atoms are trapped in a basement at the time it is sealed, how long will it tae for 90% of the original quantity of gas to decay?

6 Lecture_07_02.nb 6 Heat Transfer: Newton's Law of Cooling Newton's law of cooling states that the rate at which an object's temperature is changing at any given time is proportional to the difference between its temperature and the temperature of the surrounding medium. Let H be the temperature of an object at time t, H S the constant surrounding temperature. Then the differential equation is dh ÅÅÅÅÅÅÅÅ d t = -HH - H S L, HH0L = H 0 Find a formula for the solution to the differential equation dh ÅÅÅÅÅÅÅÅ d t = -HH - H S L, HH0L = H 0 Your cup of coffee is 185 and you decide to let it cool. After two minutes you chec the coffee and it has a temperature of 155. If the temperature of the classroom is a constant 65, how much longer will you have to wait for the coffee to cool to 105?