Population Growth The population of the world in the year

Problem 45A Newton’s Law of Cooling Newton’s law of cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and an environment into which it is introduced. This leads to an equation where the temperature f(t) of the body at time t after being introduced into an environment having constant temperature T0 is f(t) = T0 + Ce?kt, where C and k are constants. Use this result. If C = ?14.6 and k = 0.6 and t is time in hours, how long will it take a frozen pizza to thaw to 10°C in a room at 18°C?

Problem 1E What is the difference between stated interest rate and effective interest rate?

Problem 4E What is meant by the half-life of a quantity?

Problem 2E In the exponential growth or decay function y = y0ekt, what does y0 represent? What does krepresent?

Problem 5E Show that if a radioactive substance has a half-life of T, then the corresponding constant k in the exponential decay function is given by k = ?(ln 2)/T.

Problem 3E In the exponential growth or decay function, explain the circumstances that cause k to be positive or negative.

Problem 6E Show that if a radioactive substance has a half-life of T, then the corresponding exponential decay function can be written as y = y0(1/2)(t/T).

Problem 7A Effective Rate Find the effective rate corresponding to the nominal rate of interest. 4% compounded quarterly

Problem 8A Effective Rate Find the effective rate corresponding to the nominal rate of interest. 6% compounded monthly

Problem 9A Effective Rate Find the effective rate corresponding to the nominal rate of interest. 8% compounded continuously

Problem 10A Effective Rate Find the effective rate corresponding to the nominal rate of interest. 5% compounded continuously

Problem 11A Present Value Find the present value of the amount. $10,000 if interest is 6% compounded quarterly for 8 years

Problem 13A Present Value Find the present value of the amount. $7300 if interest is 5% compounded continuously for 3 years

Problem 12A Present Value Find the present value of the amount. $45,678.93 if interest is 7.2% compounded monthly for 11 months

Problem 14A Present Value Find the present value of the amount. $25,000 if interest is 4.6% compounded continuously for 8 years

Problem 15A Effective Rate Tami Dreyfus bought a television set with money borrowed from the bank at 9% interest compounded semiannually. What effective interest rate did she pay?

Problem 16A Effective Rate A firm deposits some funds in a special account at 6.2% compounded quarterly. What effective rate will they earn?

Problem 17A Effective Rate Robin Kim deposits $7500 of lottery winnings in an account paying 6% interest compounded monthly. What effective rate does the account earn?

Problem 18A Present Value Frank Steed must make a balloon payment of $20,000 in 4 years. Find the present value of the payment if it includes annual interest of 6.5% compounded monthly.

Problem 19A Present Value A company must pay a $307,000 settlement in 3 years. a. What amount must be deposited now at 6% compounded semiannually to have enough money for the settlement? ________________ b. How much interest will be earned? ________________ c. Suppose the company can deposit only $200,000 now. How much more will be needed in 3 years? ________________ d. Suppose the company can deposit $200,000 now in an account that pays interest continuously. What interest rate would they need to accumulate the entire $307,000 in 3 years?

Problem 20A Present Value A couple wants to have $40,000 in 5 years for a down payment on a new house. a. How much should they deposit today, at 6.4% compounded quarterly, to have the required amount in 5 years? ________________ b. How much interest will be earned? ________________ c. If they can deposit only $20,000 now, how much more will they need to complete the $40,000 after 5 years? ________________ d. Suppose they can deposit $20,000 now in an account that pays interest continuously. What interest rate would they need to accumulate the entire $40,000 in 5 years?

Problem 21A Interest Christine O’Brien, who is self-employed, wants to invest $60,000 in a pension plan. One investment offers 8% compounded quarterly. Another offers 7.75% compounded continuously. a. Which investment will earn the most interest in 5 years? ________________ b. How much more will the better plan earn? ________________ c. What is the effective rate in each case? ________________ d. If Ms. O’Brien chooses the plan with continuous compounding, how long will it take for her $60,000 to grow to $80,000? ________________ e. How long will it take for her $60,000 to grow to at least $80,000 if she chooses the plan with quarterly compounding? (Be careful; interest is added to the account only every quarter. See Example.) Interest Meghan Moreau has received a bonus of $25,000. She invests it in an account earning 7.2% compounded quarterly. Find how long it will take for her $25,000 investment to grow to $40,000. SOLUTION Here P = $25,000, r = 0.072, and m = 4. We also know the amount she wishes to end up with, A = $40,000. Substitute these values into the compound interest formula and solve for time, t. Note that the interest is calculated quarterly and is added only at the end of each quarter. Therefore, we need to round up to the nearest quarter. She will have $40,000 in 6.75 years. CAUTION When calculating the time it takes for an investment to grow, take into account that interest is added only at the end of each compounding period. In Example, interest is added quarterly. At the end of the second quarter of the sixth year (t = 6.5), she will have only $39,754.13, but at the end of the third quarter of that year (t = 6.75), she will have $40,469.70. If A, the amount of money we wish to end up with, is given as well as r, m, and t, then P can be found using the formula for compounded interest. Here P is the amount that should be deposited today to produce A dollars in t years. The amount P is called the present value of A dollars.

Problem 24A Sales Sales of a new model of digital camera are approximated by S(x) = 5000 ?4000e?x, where x represents the number of years that the digital camera has been on the market, and S(x)represents sales in thousands of dollars. a. Find the sales in year 0. ________________ b. When will sales reach $4,500,000? ________________ c. Find the limit on sales.

Problem 25A Population Growth The population of the world in the year 1650 was about 500 million, and in the year 2010 was 6756 million. Source: U.S. Census Bureau. a. Assuming that the population of the world grows exponentially, find the equation for the population P(t) in millions in the year t. ________________ b. Use your answer from part a to find the population of the world in the year 1. ________________ c. Is your answer to part b reasonable? What does this tell you about how the population of the world grows?

Problem 26A Giardia When a person swallows giardia cysts, stomach acids and pancreatic enzymes cause the cysts to release trophozoites, which divide every 12 hours. Source: The New York Times. a. Suppose the number of trophozoites at time t = 0 is y0. Write a function in the form y = y0ektgiving the number after t hours. ________________ b. Write the function from part a in the form y = y02f(t). ________________ c. The article cited above said that a single trophozoite can multiply to a million in just 10 days and a billion in 15 days. Verify this fact.

Problem 23A Sales Sales of a new model of compact disc player are approximated by the function S(x) = 1000 ?800e?x, where S(x) is in appropriate units and x represents the number of years the disc player has been on the market. a. Find the sales during year 0. ________________ b. In how many years will sales reach 500 units? ________________ c. Will sales ever reach 1000 units? ________________ d. Is there a limit on sales for this product? If so, what is it?

Problem 22A Interest Greg Tobin wishes to invest a $5000 bonus check into a savings account that pays 6.3% interest. Find how many years it will take for the $5000 to grow to at least $11,000 if interest is compounded a. quarterly. (Be careful; interest is added to the account only every quarter. See Example.) ________________ b. continuously. Interest Meghan Moreau has received a bonus of $25,000. She invests it in an account earning 7.2% compounded quarterly. Find how long it will take for her $25,000 investment to grow to $40,000. SOLUTION Here P = $25,000, r = 0.072, and m = 4. We also know the amount she wishes to end up with, A = $40,000. Substitute these values into the compound interest formula and solve for time, t. Note that the interest is calculated quarterly and is added only at the end of each quarter. Therefore, we need to round up to the nearest quarter. She will have $40,000 in 6.75 years. CAUTION When calculating the time it takes for an investment to grow, take into account that interest is added only at the end of each compounding period. In Example, interest is added quarterly. At the end of the second quarter of the sixth year (t = 6.5), she will have only $39,754.13, but at the end of the third quarter of that year (t = 6.75), she will have $40,469.70. If A, the amount of money we wish to end up with, is given as well as r, m, and t, then P can be found using the formula for compounded interest. Here P is the amount that should be deposited today to produce A dollars in t years. The amount P is called the present value of A dollars.

Problem 27A Growth of Bacteria A culture contains 25,000 bacteria, with the population increasing exponentially. The culture contains 40,000 bacteria after 10 hours. a. Write a function in the form y = y0ekt giving the number of bacteria after t hours. ________________ b. Write the function from part a in the form y = y0at. ________________ c. How long will it be until there are 60,000 bacteria?

Problem 29A Growth of Bacteria The growth of bacteria in food products makes it necessary to time-date some products (such as milk) so that they will be sold and consumed before the bacteria count is too high. Suppose for a certain product that the number of bacteria present is given by under certain storage conditions, where t is time in days after packing of the product and the value of f(t) is in millions. a. If the product cannot be safely eaten after the bacteria count reaches 3000 million, how long will this take? ________________ b. If t = 0 corresponds to January 1, what date should be placed on the product?

Problem 28A Decrease in Bacteria When an antibiotic is introduced into a culture of 50,000 bacteria, the number of bacteria decreases exponentially. After 9 hours, there are only 20,000 bacteria. a. Write an exponential equation to express the growth function y in terms of time t in hours. ________________ b. In how many hours will half the number of bacteria remain?

Problem 30A Cancer Research An article on cancer treatment contains the following statement: A 37% 5-year survival rate for women with ovarian cancer yields an estimated annual mortality rate of 0.1989. The authors of this article assume that the number of survivors is described by the exponential decay function given at the beginning of this section, where y is the number of survivors and k is the mortality rate. Verify that the given survival rate leads to the given mortality rate. Source: American Journal of Obstetrics and Gynecology.

Problem 33A Find the half life of the radioactive substance. See example 2 Plutonium-241;A(t) = A0e?0.053t

Problem 34A Find the half life of the radioactive substance. See example 2 Radium-226; A(t) = A0e?0.00043t

Problem 36A Half-Life The half-life of radium-226 is approximately 1620 years. a. How much of a sample weighing 4 g will remain after 100 years? ________________ b. How much time is necessary for a sample weighing 4 g to decay to 0.1 g?

Problem 35A Half-Life The half-life of plutonium-241 is approximately 13 years. a. How much of a sample weighing 4 g will remain after 100 years? ________________ b. How much time is necessary for a sample weighing 4 g to decay to 0.1 g?

Problem 38A Radioactive Decay 25 g of polonium-210 is decaying exponentially. After 50 days 19.5 g of polonium-210 is left. ________________ a. Write a function in the form y = y0ekt giving the number of grams of polonium-210 after t days. ________________ b. Write the function from part a in the form y = y0at/50. ________________ c. Use your answer from part a to find the half-life of polonium-210.

Problem 31A Chromosomal Abnormality The graph below shows how the risk of chromosomal abnormality in a child rises with the age of the mother. Source: downsyndrome.about.com. a. Read from the graph the risk of chromosomal abnormality (per 1000) at ages 20, 35, 42, and 49. ________________ b. Assuming the graph to be of the form y = Cekt, find k using t = 20 and t = 35. ________________ c. Still assuming the graph to be of the form y = Cekt, find k using t = 42 and t = 49. ________________ d. Based on your results from parts a-c, is it reasonable to assume the graph is of the form y = Cekt? Explain. ________________ e. In situations such as parts a?c, where an exponential function does not fit because different data points give different values for the growth constant k, it is often appropriate to describe the data using an equation of the form y = Cekt. Parts b and c show that n = 1 results in a smaller constant using the interval [20, 35] than using the interval [42, 49]. Repeat parts b and c using n = 2, 3, etc., until the interval 20, 35] yields a larger value of k than the interval [42, 49], and then estimate what n should be.

Problem 37A Radioactive Decay 500 g of iodine-131 is decaying exponentially. After 3 days 386 g of iodine-131 is left. a. Write a function in the form y = y0ekt giving the number of grams of iodine-131 after t days. ________________ b. Write the function from part a in the form y = y0 (386/500)f(t). ________________ c. Use your answer from part a to find the half-life of iodine-131.

Problem 32A Carbon Dating Refer to Example. A sample from a refuse deposit near the Strait of Magellan had 60% of the carbon-14 found in a contemporary living sample. How old was the sample? Half-Life Find the half-life of each radioactive substance. See Example. Carbon Dating Carbon-14 is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies, the carbon-14 disintegrates. Scientists determine the age of the remains by comparing its carbon-14 with the amount found in living plants and animals. The amount of carbon-14 present after t years is given by the exponential equation with k = ?[(ln 2)/5600]. (a) Find the half-life of carbon-14. SOLUTION Let A(t) = (1/2)A0 and k = ?[(ln 2)/5600]. The half-life is 5600 years. (b) Charcoal from an ancient fire pit on Java had 1/4 the amount of carbon-14 found in a living sample of wood of the same size. Estimate the age of the charcoal. SOLUTION Let A (t) = (1/4)A0 and k = ?[(ln 2)/5600]. The charcoal is about 11,200 years old. By following the steps in Example, we get the general equation giving the half-life T in terms of the decay constant k as For example, the decay constant for potassium-40, where t is in billions of years, is approximately ?0.5545 so its half-life is We can rewrite the growth and decay function as y = y0 ekt = y0(ek)t = y0 a t, where a = ek. This is sometimes a helpful way to look at an exponential growth or decay function.

Problem 39A Nuclear Energy Nuclear energy derived from radioactive isotopes can be used to supply power to space vehicles. The output of the radioactive power supply for a certain satellite is given by the function y = 40e?0004t, where y is in watts and t is the time in days. a. How much power will be available at the end of 180 days? ________________ b. How long will it take for the amount of power to be half of its original strength? ________________ c. Will the power ever be completely gone? Explain.

Problem 40A Botany A group of Tasmanian botanists have claimed that a King’s holly shrub, the only one of its species in the world, is also the oldest living plant. Using carbon-14 dating of charcoal found along with fossilized leaf fragments, they arrived at an age of 43,000 years for the plant, whose exact location in southwest Tasmania is being kept a secret. What percent of the original carbon-14 in the charcoal was present? Source: Science.

Problem 41A Decay of Radioactivity A large cloud of radioactive debris from a nuclear explosion has floated over the Pacific Northwest, contaminating much of the hay supply. Consequently, farmers in the area are concerned that the cows who eat this hay will give contaminated milk. (The tolerance level for radioactive iodine in milk is 0.) The percent of the initial amount of radioactive iodine still present in the hay after t days is approximated by P(t), which is given by the mathematical model P(t) = 100e?0.1t. a. Find the percent remaining after 4 days. ________________ b. Find the percent remaining after 10 days. ________________ c. Some scientists feel that the hay is safe after the percent of radioactive iodine has declined to 10% of the original amount. Solve the equation 10 = 100e?0.1t to find the number of days before the hay may be used. ________________ d. Other scientists believe that the hay is not safe until the level of radioactive iodine has declined to only 1% of the original level. Find the number of days that this would take.

Problem 42A Chemical Dissolution The amount of chemical that will dissolve in a solution increases exponentially as the temperature is increased. At 0°C, 10 g of the chemical dissolves, and at 10°C, 11 g dissolves. a. Write an equation to express the amount of chemical dissolved, y, in terms of temperature, t, in degrees Celsius. ________________ b. At what temperature will 15 g dissolve?

Problem 44A Newton’s Law of Cooling Newton’s law of cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and an environment into which it is introduced. This leads to an equation where the temperature f(t) of the body at time t after being introduced into an environment having constant temperature T0 is f(t) = T0 + Ce?kt, where C and k are constants. Use this result. If C = 100, k = 0.1, and t is time in minutes, how long will it take a hot cup of coffee to cool to a temperature of 25°C in a room at 20°C?

Problem 43A Newton’s Law of Cooling Newton’s law of cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and an environment into which it is introduced. This leads to an equation where the temperature f(t) of the body at time t after being introduced into an environment having constant temperature T0 is f(t) = T0 + Ce?kt, where C and k are constants. Use this result. Find the temperature of an object when t = 9 if T0 = 18, C = 5, and k = 0.6.