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Q 3.2-23E

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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 101
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

In Problems 23–27, assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The half-life of a radioactive substance is the time it takes for one-half of the substance to disintegrate. If initially there are 50 g of a radioactive substance and after 3 days there are only 10 g remaining, what percentage of the original amount remains after 4 days?

After 4 days, the remaining radioactive substance will be 11.7% of the original amount.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Analyzing the given statement

Given that the rate of decay of a radioactive substance is directly proportional to the amount of the substance present. Let the present amount of the radioactive substance be N.

Therefore, dNdt∝N

Given that there are 50g of a radioactive substance and after 3 days there are only 10g remain. We have to find the mass of the substance, remaining after 4 days, and its percentage of the original amount.

Step 2: Determining the formula with the help of the given proportionality relation, to solve the question 

Given,

dNdt∝NdNdt=-λN

where, λ is the constant of proportionality.

dNN=-λ∫dNN=-λ∫dt lnN=-λt+lnN0

where, In N0 is an arbitrary constant.

lnN-lnN0=-λt lnNN0=-λt NN0=e-λt N=N0e-λt······1

One will use this formula to solve the question.

Step 3: Using the formula obtained in the step 2, we will find the value of λ

Let the initial amount of the radioactive substance be i.e., N0= 50g and given that the remaining amount of radioactive substance after 3 days is 10g i.e.,

t = 3 days and N = 10g

Now, from the equation (1),

10=50e-3λ 15=e-3λ e3λ=5 3λ=ln 5 λ=ln 53 λ=0.5365

One will use this value of λ in the next step to find the value of the mass of the remaining radioactive substance after 4days.

Step 4: Finding the mass of the remaining radioactive substance after 4 days

Now we will find the mass of the remaining radioactive substance after 4 days.

For this, let N be the mass to be found,

N0=50 g

Time, t = 4 days

(From Step 3)

Using the equation (1),

N=N0e-λtN=50·e-0.53654N=5.847 g

Hence, the mass of the remaining radioactive substance after 4 days is 5.847 g.

Step 5: Determining what percentage of the original amount remains after 4 days

The mass of the remaining radioactive substance after 4 days is 5.847 g

Therefore,

Percentage of remaining mass

=5.847N0×100=5.84750×100=11.7%

Thus, the percentage of the original amount that remains after 4 days is 11.7%.

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