Question

a scientist begins with 275 grams of a radioactive substance. after 260 minutes, the sample has decayed to 30 grams. to the nearest hundredth of a minute, what is the half - life of this substance?

Explanation:

Step1: Write the radioactive - decay formula

The radioactive - decay formula is $A = A_0e^{-kt}$, where $A$ is the amount of the substance at time $t$, $A_0$ is the initial amount of the substance, $k$ is the decay constant, and $t$ is the time. We know that $A_0 = 275$ grams, $A = 30$ grams, and $t = 260$ minutes. Substitute these values into the formula:
$30=275e^{-260k}$

Step2: Solve for $k$

First, divide both sides of the equation by 275:
$\frac{30}{275}=e^{-260k}$
$\frac{6}{55}=e^{-260k}$
Take the natural logarithm of both sides: $\ln(\frac{6}{55})=\ln(e^{-260k})$.
Since $\ln(e^{-260k})=-260k$, we have $k =-\frac{\ln(\frac{6}{55})}{260}$.
$\ln(\frac{6}{55})=\ln(6)-\ln(55)\approx1.79176 - 4.00733=-2.21557$
$k=\frac{2.21557}{260}\approx0.0085214$

Step3: Use the half - life formula

The half - life formula is $t_{1/2}=\frac{\ln(2)}{k}$.
Substitute $k\approx0.0085214$ into the formula:
$t_{1/2}=\frac{\ln(2)}{0.0085214}$
Since $\ln(2)\approx0.69315$, then $t_{1/2}=\frac{0.69315}{0.0085214}\approx81.34$ minutes

Answer:

$81.34$