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Exponential Growth or Decay Function In many situations that occur in ecology, biology, economics, and the social sciences, a quantity changes at a rate proportional to the amount present. In such cases the amount present at time t is a special function of t called an exponential growth or decay function.

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Exponential Growth or Decay Function Let y 0 be the amount or number present at time t = 0. Then, under certain conditions, the amount present at any time t is modeled by where k is a constant.

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Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Solution a. Find an exponential function that gives the amount of carbon dioxide y in year t. Recall that the graph of the data points showed exponential growth, so the equation will take the form y = y 0 e kt. We must find the values of y 0 and k. The data began with the year 1990, so to simplify our work we let 1990 correspond to t = 0, 1991 correspond to t = 1, and so on. Since y 0 is the initial amount, y 0 = 353 in 1990, that is, when t = 0.

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Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Solution a. Find an exponential function that gives the amount of carbon dioxide y in year t. Thus the equation is From the last pair of values in the table, we know that in 2275 the carbon dioxide level is expected to be 2000 ppm. The year 2275 corresponds to 2275 – 1990 = 285. Substitute 2000 for y and 285 for t and solve for k.

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Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Divide by 353. Take logarithms on both sides. In e x = x Solution

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Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Multiply by ; rewrite. Use a calculator. A function that models the data is Solution

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Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE Solution b. Let y = 2(280) = 560 in y = 353 e t, and find t. Divide by 353. Take logarithms on both sides. Estimate the year when future levels of carbon dioxide will double the 1951 level of 280 ppm.

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Example 1 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL THE INCREASE OF CARBON DIOXIDE In e x = x Multiply by ; Rewrite. Use a calculator. Since t = 0 corresponds to 1990, the 1951 carbon dioxide level will double in the 75th year after 1990, or during 2065.

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Example 2 FINDING DOUBLING TIME FOR MONEY How long will it take for the money in an account that is compounded continuously at 3% interest to double? Solution Continuous compounding formula Let A = 2P and r =.03 Divide by P Take logarithms on both sides.

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Example 2 FINDING DOUBLING TIME FOR MONEY How long will it take for the money in an account that is compounded continuously at 3% interest to double? Solution Divide by.03 In e x = x Use a calculator. It will take about 23 yr for the amount to double.

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Example 3 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL POPULATION GROWTH According to the U.S. Census Bureau, the world population reached 6 billion people on July 18, 1999, and was growing exponentially. By the end of 2000, the population had grown to billion. The projected world population (in billions of people) t years after 2000, is given by the function defined by

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Example 3 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL POPULATION GROWTH Based on this model, what will the world population be in 2010? Solution a. Since t = 0 represents the year 2000, in 2010, t would be 2010 – 2000 = 10 yr. We must find (t) when t is 10. Let t = 10. According to the model, the population will be billion at the end of 2010.

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Example 3 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL POPULATION GROWTH Solution b. Let t = 7. In what year will the world population reach 7 billion? Divide by Take logarithms on both sides

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Example 3 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL POPULATION GROWTH Solution b. In what year will the world population reach 7 billion? In e x = x Divide by.0126; rewrite. Use a calculator. World population will reach 7 billion 11.2 yr after 2000, during the year 2011.

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Example 4 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL RADIOACTIVE DECAY If 600 g of a radioactive substance are present initially and 3 yr later only 300 g remain, how much of the substance will be present after 6 yr? Solution To express the situation as an exponential equation we use the given values to first find y 0 and then find k.

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Example 4 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL RADIOACTIVE DECAY If 600 g of a radioactive substance are present initially and 3 yr later only 300 g remain, how much of the substance will be present after 6 yr? Solution Let y = 600 and t = 0. e 0 = 1 Thus, y 0 = 600 and the exponential equation y = y 0 e kt becomes

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Example 4 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL RADIOACTIVE DECAY Let y = 300 and t = 3. Divide by 600. Now solve this exponential equation for k. Take logarithms on both sides. Solution

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Example 4 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL RADIOACTIVE DECAY Divide by 3. In e x = x Use a calculator. Solution

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Example 4 DETERMINING AN EXPONENTIAL FUNCTION TO MODEL RADIOACTIVE DECAY Thus, the exponential decay equation is y = 600 e –.231t. To find the amount present after 6 yr, let t = 6. After 6 yr, about 150 g of the substance will remain. Solution

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Half Life Analogous to the idea of doubling time is half-life, the amount of time that it takes for a quantity that decays exponentially to become half its initial amount.

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Example 5 SOLVING A CARBON DATING PROBLEM Carbon 14, also known as radiocarbon, is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. Scientists can determine the age of the remains by comparing the amount of radiocarbon with the amount present in living plants and animals. This technique is called carbon dating. The amount of radiocarbon present after t years is given by where y 0 is the amount present in living plants and animals.

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Example 5 SOLVING A CARBON DATING PROBLEM Find the half-life. Solution a. If y 0 is the amount of radiocarbon present in a living thing, then ½ y 0 is half this initial amount. Thus, we substitute and solve the given equation for t. Let y = ½ y 0.

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Example 5 SOLVING A CARBON DATING PROBLEM Find the half-life. Solution a. Let y = ½ y 0. Divide by y 0. Take logarithms on both sides.

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Example 5 SOLVING A CARBON DATING PROBLEM Find the half-life. Solution a. Divide by – In e x = x Use a calculator. The half-life is about 5700 yr.

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Example 5 SOLVING A CARBON DATING PROBLEM Solution b. Charcoal from an ancient fire pit on Java contained ¼ the carbon 14 of a living sample of the same size. Estimate the age of the charcoal. Solve again for t, this time letting the amount y = ¼ y 0. Given equation. Let y = ¼ y 0.

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Example 5 SOLVING A CARBON DATING PROBLEM Divide by y 0. Take logarithms on both sides In e x = x; divide by Use a calculator. The charcoal is about 11,400 yr old. Solution

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Example 6 MODELING NEWTONS LAW OF COOLING Newtons law of cooling says that the rate at which a body cools is proportional to the difference C in temperature between the body and the environment around it. The temperature (t) of the body at time t in appropriate units after being introduced into an environment having constant temperature T 0 is where C and k are constants.

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Example 6 MODELING NEWTONS LAW OF COOLING A pot of coffee with a temperature of 100°C is set down in a room with a temperature of 20°C. The coffee cools to 60°C after 1 hr. Write an equation to model the data. a. Solution We must find values for C and k in the formula for cooling. From the given information, when t = 0, T 0 = 20, and the temperature of the coffee is (0) = 100. Also, when t = 1, (1) = 60. Substitute the first pair of values into the equation along with T 0 =20.

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Example 6 MODELING NEWTONS LAW OF COOLING Write an equation to model the data. a. Solution Given formula Let t = 0, (0) = 100, and T 0 = 20. e 0 = 1 Subtract 20. Thus,

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Example 6 MODELING NEWTONS LAW OF COOLING Given formula Let t = 1, (1) = 60. Subtract 20. Now use the remaining pair of values in this equation to find k. Divide by 80. Take logarithms on both sides. Solution

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Example 6 MODELING NEWTONS LAW OF COOLING Thus, In e x = x Solution Now use the remaining pair of values in this equation to find k.

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To find the temperature after ½ hr, let t = ½ in the model from part (a). Example 6 MODELING NEWTONS LAW OF COOLING b. Solution Model from part (a) Let t = ½. Find the temperature after half an hour.

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To find how long it will take for the coffee to cool to 50°C, let (t) = 50. Example 6 MODELING NEWTONS LAW OF COOLING c. Solution Let (t) = 50. Subtract 20. How long will it take for the coffee to cool to 50°C? Divide by 80.

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To find how long it will take for the coffee to cool to 50°C, let (t) = 50. Example 6 MODELING NEWTONS LAW OF COOLING c. Solution How long will it take for the coffee to cool to 50°C? Take logarithms on both sides. In e x = x

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