Concept explainers
In Exercises 6—10, we consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law:
The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used.
7. The half-life of a radioactive isotope is the amount of time it takes for a quantity of radioactive material to decay to one-half of its original amount.
(a) The half-life of Carbon 14 (C-14) is 5230 years. Determine the decay-rate pa-rameter
(b) The half-life of Iodine 131 (I-131) is 8 days. Determine the decay-rate param-eter for I-131.
(c) What are the units of the decay-rate parameters in parts (a) and (b)?
(d) To determine the half-life of an isotope, we could start with atoms of the isotope and measure the amount of time it takes 500 of them to decay, or we could start with 10,000 atoms of the isotope and measure the amount of time it takes 5000 of them to decay. Will we get the same answer? Why?
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Chapter 1 Solutions
DIFFERENTIAL EQUATIONS-ACCESS
- The table shows the mid-year populations (in millions) of five countries in 2015 and the projected populations (in millions) for the year 2025. (a) Find the exponential growth or decay model y=aebt or y=aebt for the population of each country by letting t=15 correspond to 2015. Use the model to predict the population of each country in 2035. (b) You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation y=aebt gives the growth rate? Discuss the relationship between the different growth rates and the magnitude of the constant.arrow_forwardSuppose that the initial mass of radioactive substance is m0 and the half-life of the substance is h. Let m(t) be the mass remaining at time t. a What is meant by the half-life h? b Write a formula for m(t) in terms of the half-life h. c Write a formula for the relative decay rate r in terms of the half-life h. d Write a formula for m(t) in terms of the relative decay rate r.arrow_forwardSodium-24 is a radioactive isotope that is used in diagnosing circulatory disease. It decays into stable isotopes of sodium at a rate of 4.73% per hour. a. Technicians inject a quantity of sodium-24 into a patient's bloodstream. Write a formula for the amount of sodium-24 present in the bloodstream as a function of time. Use A for the initial amount of sodium-24. S(t)= ________?_________ b. How long will it take for 75% of the isotope to decay? Answer exactly or round to 3 decimal places. ______?_______ hoursarrow_forward
- Exercise 2.1.6 A body is found at a certain time and has a temperature of 92.6 degrees Fahren- heit in an environment with ambient temperature 70 degrees. Two hours later the body has a temperature of 90 degrees Fahrenheit. The flu was going around and it was believed the victim was on her way to the drugstore because her roommate said she had a temperature of 102.4 degrees Fahrenheit. If Newton's law of cooling holds, estimate when the person died, relative to the time the body was found. Hint: call the time the body is found t = 0 and solve u' (t) = -k(u(t) -A) with A = 70 and initial condition u(0) = 92.6. Then use u(2) = 90 to find k, and from that figure out at what time u(t) equaled 102.4.arrow_forwardWhich function represents a decaying element that starts with 75% of its original quantity, with a half life of 5 hours, When P represents the percentage of quantity remaining and t represents the number of hours that have passed? a. P=0.75 b. 75=P 12 5t C. p = 0.75 0.75(1) d. e. 5t P=75 P = 75 (1) 1 P = 75(1) Xarrow_forward1. The growth of a population of a fishpond is proportional to its current population. A pond initially has 500 fish. After six months it has doubled in population. The owner will only allow fishing in his pond after it has grown to a population of 10,000. When will the owner allow fishing?arrow_forward
- 1. The growth of a population of a fishpond is proportional to its current population. A pond initially has 500 fish. After six months it has doubled in population. The owner will only allow fishing in his pond after it has grown to a population of 10,000. When will the owner allow fishing?arrow_forward#12arrow_forwardnèstion 2 A 13-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by -0.089r m(t) = 13e %3D where m(t) is measured in grams. After how many days is there only 10 g remaining? O a. 3 days O b.6 days c. 2 days d. 4 days e. 5 days A Moving to the next question prevents changes to this answer.arrow_forward
- #13A and #13Barrow_forward10. In 2013, the population of a city was 715,000, with an annual growth rate of 0.013% a) Find a mathematical model that describes the population t years after 2013. The model is P b) if the annual rate of increase remains the same, use this model to predict the population of the country in the year 2021. Round to the nearest whole humber...arrow_forward
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