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zheka24 [161]
2 years ago
9

The exponential decay function A = A0(1/2)^t/P can be used to determine the amount A, of a radioactive substance present at time

t, if A0 represents the initial amount and P represents the half-life of the substance.
If a substance loses 70% of its radioactivity in 500 days, determine the period of the half-life.
Show your work.
Mathematics
1 answer:
amid [387]2 years ago
4 0

Answer:

The half-life of the substance is about 288 days.

Step-by-step explanation:

The exponential decay function:

\displaystyle A=A_0\left(\frac{1}{2}\right)^{t/P}

Can determine the amount <em>A</em> of a radioactive substance present at time <em>t. A₀ </em>represents the initial amount and <em>P</em> is the half-life of the substance.

We are given that a substance loses 70% of its radioactivity in 500 days, and we want to determine the period of the half-life.

In other words, we want to determine <em>P. </em>

Since the substance has lost 70% of its radioactivity, it will have only 30% of its original amount. This occured in 500 days. Therefore, <em>A</em> = 0.3<em>A₀</em> when <em>t</em> = 500 (days). Substitute:

\displaystyle 0.3A_0=A_0\left(\frac{1}{2}\right)^{500/P}

Divide both sides by <em>A₀:</em>

\displaystyle 0.3=\left(\frac{1}{2}\right)^{500/P}

We can take the natural log of both sides:

\displaystyle \ln(0.3)=\ln\left(\left(\frac{1}{2}\right)^{500/P}\right)

Using logarithmic properties:

\displaystyle \ln(0.3)=\frac{500}{P}\left(\ln\left(\frac{1}{2}\right)\right)

So:

\displaystyle \frac{500}{P}=\frac{\ln(0.3)}{\ln(0.5)}

Take the reciprocal of both sides:

\displaystyle \frac{P}{500}=\displaystyle \frac{\ln(0.5)}{\ln(0.3)}

Use a calculator:

\displaystyle P=\frac{500\ln(0.5)}{\ln(0.3)}\approx287.86

The half-life of the substance is about 288 days.

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