   Chapter 11.2, Problem 46E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem

# Radioactive Decay The rate of decomposition of radioactive einsteinium is proportional to the amount present at any time. The half-life of radioactive einsteinium is 276 days. After 100 days, 0.5 gram remains. What was the initial amount?

To determine

To calculate: The initial amount after 100 days, 0.5 grams remain.

Explanation

Given information:

The half-life of radioactive einsteinium is 276 years.

The rate of decomposition of radioactive einsteinium is proportional to the amount present at any time.

Formula Used:

Integration formula:

1ydy=ln|y|,kdt=kt+c

Calculation:

Let y represent the mass of the einsteinium and t represent the time in years.

According to question,

The rate of decomposition of radioactive einsteinium is proportional to the amount present at any time.

Therefore,

dydt=ky

Consider the differential equation,

dydt=ky

Separate the variable by grouping the t variable on side and the y variable on other side,

Therefore,

dydt=kydyy=kdt

Integrate each side of the above differential equation,

Therefore,

dyy=kdt

Apply, integration formula 1ydy=ln|y|,kdt=kt+c in dyy=kdt

Therefore,

dyy=kdtln|y|=

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