Asample of a radioactive substance decayed to 93.5% of its original amount after a year. (round your answers to two decimal places.) (a) what is the half-life of the substance? yr (b) how long would it take the sample to decay to 90% of its original amount? yr

Using an exponential equation, we have that:

a) The half-life of the substance is of 10.31 years.

b) It would take 1.57 years for the sample to decay to 90% of its original amount.

The equation for the amount of substance after t years is given by:

In which:

93.5% of its original amount after a year.

This means that , and this is used to find k.

Thus:

Item a:

The half-life is t for which:

Then

The half-life of the substance is of 10.31 years.

Item b:

This is t for which:

Then

It would take 1.57 years for the sample to decay to 90% of its original amount.

A similar problem is given at link

a). Half life period = 10.31 years

b). Time taken to decay to 90% of the original amount = 1.57 years

Step-by-step explanation:

Decay of a radioactive element is represented by the exponential function.

Where = Amount of the element after decay

= Initial amount

k = Decay constant

t = time or duration

Substance decayed 93.5% after one year.

Therefore, 93.5% of =

Take the log on both the sides of the equation

-0.067208 = -k

k = 0.067208

(a). For half life of the substance

\frac{1}{2}=e^{-0.067208t}  

Take the log on each side

0.693147 = 0.067208t

t =

 = 10.31 years

(b). Time taken to decay to 90% of the original amount

Take log on both the sides of the equation

-0.1053605 = -0.067208t

t =

 = 1.57 years

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