   Chapter 11, Problem 29RE ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Radioactive decay A breeder reactor converts stable uranium-238 into the isotope plutonium-239. The decay of this isotope is given by A ( t ) =   A 0 e − 0.00002876 t where A(t) is the amount of isotope at time t, in years, and A 0 is the original amount. This isotope has a half- life of 24,101 years (that is, half of it will decay away in 24,101 years).(a) At what rate is A(t) decaying at t = 24,101 years?(b) At what rate is A(t) decaying after 1 year?(c) Is the rate of decay at its half-life greater or less than after 1 year?

(a)

To determine

To calculate: The rate at which A(t) is decaying at t=24,101 if a breeder reactor converts stable uranium-238 into the isotope plutonium-239. The decay of this isotope is given by A(t)=A0e0.00002876t where A(t) is the amount of isotope at time t, in years and A0 is the original amount. The isotope has a half-life of 24,101 years.

Explanation

Given Information:

The provided expression is A(t)=A0e0.00002876t,

Where A(t) is the amount of isotope at time t, in years and A0 is the original amount. The isotope has a half-life of 24,101 years.

Formula Used:

The product rule, if two functions are given in the form f(x).g(x), then the derivative is given as:

ddx(fg)=fdgdx+gdfdx

The first derivative represent the rate.

Calculation:

The provided expression is A(t)=A0e0.00002876t,

Differentiate on both the sides with respect to t by the product rule,

(b)

To determine

To calculate: The rate at which A(t) is decaying at t=1year if a breeder reactor converts stable uranium-238 into the isotope plutonium-239. The decay of this isotope is given by A(t)=A0e0.00002876t where A(t) is the amount of isotope at time t, in years and A0 is the original amount. The isotope has a half-life of 24,101 years.

(c)

To determine

To calculate: The rate of decay at its half-life greater or less than after 1 year if a breeder reactor converts stable uranium-238 into the isotope plutonium-239. The decay of this isotope is given by A(t)=A0e0.00002876t where A(t) is the amount of isotope at time t, in years and A0 is the original amount. The isotope has a half-life of 24,101 years.

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