   Chapter 12.5, Problem 48E ### 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 certain radioactive substance has a half-life of 50 hours. Find how long it will take for 90 % of the radioactivity to be dissipated if the amount of material x satisfies d x d t = k x ( t  in hours,  k  constant )

To determine

To calculate: The time require to dissipate the radioactive substance by 90% when the amount of radioactive substance x satisfies dxdt=kx.

Explanation

Given information:

The half-life of a certain radioactive substance is 50 hours and the amount of radioactive substance x dxdt=kx.

Formula used:

Suppose that a radioactive substance has a half-life of 50 hours. it will take for 90% of the radioactive to be dissipated if the amount of material satisfies

dxdt=kx

When a differential equation can be equivalently expressed in the form:

g(y)dy=f(x)dx

Calculation:

Consider the amount of radioactive substance x dxdt=kx.

Rewrite the provided differential equation,

dxx=kdt

Integrate both side to the differential equation,

dxx=kdtlnx=kt++Kx=ekt+K=Cekt

Further solve as:

dxx=kdtx=C

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