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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

Fossil dating Radioactive beryllium is sometimes used to date fossils found in deep-sea sediment. The amount of radioactive material x satisfies

d x d t = k x

Suppose that 10 units of beryllium are present in a living organism and that the half-life of beryllium is 4.6 million years. Find the age of a fossil if 20 % of the original radioactivity is present when the fossil is discovered.

To determine

To calculate: The age of fossil found in deep sea sediments if the amount of the radioactive beryllium material x satisfies dxdt=kx where t is the time in years and k is any constant.

Explanation

Given Information:

The rate of amount of radioactive material x at time t is directly proportional to the instant amount present is given as, dxdt=kx and the half-life of the radioactive beryllium is 4.6millionyears and also, the present amount of the beryllium present in the living organism is 10 units.

Formula used:

The logarithmic rule of integrals, 1xdx=ln|x|+C where x0.

The natural logarithm property,

logab=yb=ay.

Calculation:

Consider the rate of the amount of the radioactive material x satisfies, dxdt=kx.

Integrate both sides to the provided rate of amount of the radioactive material to get,

dxx=kdt

Now, use the logarithmic rule of integrals to get,

ln|x|=kt+C

Now, apply the property of logarithms, logab=yb=ay.

x=ekt+Cx=ekteCx=Kekt

Here, K is another constant

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