Exponential Growth and Decay The half-life of Plutonium-239 is 24,000 years. If 10 grams are present now, how long will it take until only 10% of the original sample remains? Round your answer to the nearest 10,000th. more? http://math.ucsd.edu/~wgarner/math4c/textbook/chapter4/expgrowthdecay.htm At the start of an experiment, there are 100 bacteria. If the bacteria follow an exponential growth pattern with rate k = 0.02, (a) what will be the population after 5 hours? (b) how long will it take for the population to double? Exponential Decay The value of an IPOD decreases at 35% per year. This can be modeled by the function

y = 500 · 0.65x

Make a table of values to find the value of your IPOD for the domain

{0,1,2,3,4,5,6,7,8, 9,10}

Graph your function Exponetial Growth vs. Exponetial Decay Sample Problems Sample problems Exponential Growth Growth by doubling: One of the most common examples of exponential growth deals with bacteria. Bacteria can multiply at an alarming rate when each bacteria splits into two new cells, thus doubling. For example, if we start with only one bacteria which can double every hour, by the end of one day we will have over 16 million bacteria. y = a∙bx

b > 1

getting bigger Exponential Decay y = a∙bx

b <1

getting smaller Growth or Decay? y = 5 ·2 ^x y = 4 ·0.87 ^x y = 10 ·0.02 ^x y = 6 ·1.5 ^x Growth Growth Decay Decay Exponential growth is the increase in a quantity according to the law for a parameter and constant (the analog of the decay constant), where is the exponential function and is the initial value. Exponential growth is common in physical processes such as population growth in the absence of predators or resource restrictions (where a slightly more general form is known as the law of growth). Exponential growth also occurs as the limit of discrete processes such as compound interest. Exponential Growth Exponential decay is the decrease in a quantity according to the law

for a parameter and constant (known as the decay constant), where is the exponential function and is the initial value. Exponential decay is common in physical processes such as radioactive decay, cooling in a draft (i.e., by forced convection), and so on. Exponential decay is described by the first-order ordinary differential equation Exponential Decay add text Math IV Project Growth In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after 1985. How many cell phone subscribers were in Centerville in 1994? (Don't consider a fractional part of a person.) graph it By looking at the pattern, we see that the growth in this situation can be represented as a function:

Will our formula show this same function? y=2^x amount doubles, the rate of increase is 100%. Decay by half-life: The pesticide DDT was widely used in the United States until its ban in 1972. DDT is toxic to a wide range of animals and aquatic life, and is suspected to cause cancer in humans. The half-life of DDT can be 15 or more years. Half-life is the amount of time it takes for half of the amount of a substance to decay. Scientists and environmentalists worry about such substances because these hazardous materials continue to be dangerous for many years after their disposal.

For this example, we will set the half-life of the pesticide DDT to be 15 years. Exponential growth and decay are mathematical changes. The rate of the change continues to either increase or decrease as time passes. In exponential growth, the rate of change increases over time - the rate of the growth becomes faster as time passes. In exponential decay, the rate of change decreases over time - the rate of the decay becomes slower as time passes. Since the rate of change is not constant (the same) across the entire graph, these functions are not straight lines. Solution: For (a) we are asked to find the population N at t = 5. To solve this question, we shall use the growth formula, . We are told that , k = 0.02 and that t = 5. Plugging this information in, we have the following: In (b), we are asked to determine the time that the population doubles, ie reaches 200. Again, we use the same formula, but this time we are solving for t. We have: So, the bacteria population will double in about 34.7 hours. Solution: We can use the half-life formula to find the decay rate k. We know that t = 24,000 years. Plugging into the half-life formula, we have: Now, we need to find when only 10% (1 gram) remains. Plugging into the exponential decay formula, we get the following: Notice that we rounded our answer. If you used the exact value for k, your answer would be around 79,700 years, but if you used 0.000029, your answer would be around 79,400 years. Because of this variation, only rounding to the nearest ten thousandth will yield the same answer. the end :) presented to:

leonides e. bulalayao

math iv teacher submitted by:

rico neil m. quierrez

ted michael s. sanchez

merrel hope c. santos

patrick luis dc. de guzman

jim louisr r.justo

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# Exponential Growth and Decay

Math IV Project of Quierrez, Justo, Sanchez, de Guzman and Santos

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