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.
9. Engineers and scientists often measure the rate of decay of an exponentially decaying quantity using its time constant. The time constant
(a) How are the time constant
(b) Express the time constant in terms of the half-life.
(c) What are the time constants for Carbon 14 and Iodine 131?
(d) Given an exponentially decaying quantity r(t) with initial value r0= r(0), show that its time constant is the time at which the tangent line to the graph of r(t)/r0at (0, l) crosses the t-axis. [Hint: Start by sketching the graph of r(t)/r0and the line tangent to the graph at (0, 1).]
(e) It is often said that an exponentially decaying quantity reaches its steady state in five time constants, that is, at t =5
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Differential Equations
- 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_forwardWhat is the carrying capacity for a population modeled by the logistic equation P(t)=250,0001+499e0.45t ? initial population for the model?arrow_forwardWhat is the y -intercept of the logistic growth model y=c1+aerx ? Show the steps for calculation. What does this point tell us about the population?arrow_forward
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