   Chapter 3.8, Problem 9E

Chapter
Section
Textbook Problem

The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample.(a) Find the mass that remains after t years.(b) How much of the sample remains after 100 years?(c) After how long will only 1 mg remain?

(a)

To determine

To find: The mass that remains after t years.

Explanation

Given:

Cesium-137 has a half of 30 days and y(0)=100.

Theorem used:

“The only solutions of the differential equation dydt=ky are the exponential functions

y(t)=y(0)ekt.”

Calculation:

Let y(t) be the mass that remains after t years.

By using theorem as stated above,

y(t)=y(0)ekt (1)

Substitute y(t)=50,t=30 and y(0)=100 in equation (1),

50=100e30k50100=ek12=ek

Take natural logarithm on both sides,

ln12=ln

(b)

To determine

To find: The sample remains after 100 years.

(c)

To determine

To find: The number of years after which the amount of sample left would be 1mg.

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