Chapter 4.3, Problem 15E

Reminder Round all answers to two decimal places unless otherwise indicated.

An Inappropriate Linear Model for Radioactive Decay This is a continuation of Exercise 14. Physicists have established that radioactive substances display constant percentage decay, and thus radioactive decay is appropriately modeled exponentially. This exercise is designed to show how using data without an understanding of the phenomenon that generated them can lead to inaccurate conclusions.

a. Plot the data points from Exercise 14. Do they appear almost to fall on a straight line?

b. Find the equation of the regression line and add its graph to the one you made in part a.

c. If you used the regression line as a model for decay of ${}^{\text{239}}\text{U}$, how long would it take for the Initial 1 gram to decay to half that amount? Compare this with your answer to part d of Exercise 14.

d. The linear model represented by the regression line makes an absurd prediction ‘concerning the amount of uranium Uranium-239 remaining after 1 hour. What is this prediction?

14. The half life of ${}^{239}U$ Uranium-239 is an unstable isotope of uranium that decays rapidly. In order to determine the rate of decay, 1 gram of ${}^{239}U$ was placed in a container, and the amount remaining was measured at 1-minute intervals and recorded in the table below.

Time, in minutes	Grams remaining
$0$	$1$
$1$	$0.971$
$2$	$0.943$
$3$	$0.916$
$4$	$0.889$
$5$	$0.863$

a. Show that these are exponential data and find an exponential model (For this problem, round all your answers to three decimal places.)

b. What is the percentage decay rate each minute? What does this number mean in practical terms?

c. Use functional notation to express the amount remaining after 10 minutes and then calculate the value.

d. What is the half life of ${}^{\text{239}}\text{U}$?