Section 3.8 Exponential Growth and Decay

Exponential Growth: (C is the initial value/population) The rate of change of y with respect to t is proportional to its size y(t) at any time. k is our proportionality constant

Example Use the fact that the world population was 2560 million in 1950 and 3040 million in 1960 to model the population of the world in the second half of the 20th century. (Assume that the growth rate is proportional to population size.) What is the relative growth rate? Use the model to predict the population in 1993 and 2020.

Example Radioactive Decay The half-life of radium-226 is 1590 years. A sample of radium-226 has a mass of 100 mg. Find a formula for the mass of the sample that remains after t years. Find the mass after 1000 years correct to the nearest milligram. When will the mass be reduced to 30 mg?

Newton’s Law of Cooling Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. (This law also applies to warming.):

Example A bottle of soda pop at room temperature (720 F) is placed in a refrigerator where the temperature is 440 F. After half an hour the soda pop has cooled to 610 F. What is the temperature of the soda pop after another half hour? How long does it take for the soda pop to cool to 500 F? T(30) = 540 F