   Chapter 12, Problem 21T ### 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

# A certain radioactive material has a half-life of 100 days. If the amount of material present, x, satisfies d x d t = k x where t is in days and k is constant, how long will it take for 90% of the radioactivity to dissipate?

To determine

To calculate: The time it takes for 90% of the radioactivity to dissipate if a certain radioactive material has a half-life of 100 days and the amount of material present, x, satisfies dxdt=kx.

Explanation

Given Information:

A certain radioactive material has a half-life of 100 days and the amount of material present, x, satisfies,

dxdt=kx.

Where t is in days and k is constant.

Also, initially 100g of this material is present.

Formula used:

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

g(y)dy=f(x)dx

Then the equation is separable.

The solution of a separable differential equation is obtained by integrating both sides of the equation after the variables have been separated.

According to the logarithmic rule of integrals,

1xdx=ln|x|+C

Calculation:

As it is provided that a certain radioactive material has a half-life of 100 days and the amount of material present, x, satisfies,

dxdt=kx

Where t is in days and k is constant

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