College

Element X decays radioactively with a half-life of 11 minutes. If there are 300 grams of Element X, how long, to the nearest tenth of a minute, would it take for the element to decay to 80 grams?

Answer :

In this problem we have a decay exponential function of the form

[tex]y=a(b)^x[/tex]

where

a=300 g

so

[tex]y=300(b)^x[/tex]

For x=11 min, y=300/2=150 g

substitute

[tex]\begin{gathered} 150=300(b)^{(11)} \\ solve\text{ for b} \\ b^{(11)}=\frac{150}{300} \\ b=\sqrt[11]{\frac{1}{2}} \\ \\ b=(0.5)^{\frac{1}{11}} \end{gathered}[/tex]

substitute in the equation

[tex]\begin{gathered} y=300(0.5^{(\frac{1}{11})})^x \\ y=300(0.5^)^{\frac{x}{11}} \end{gathered}[/tex]

For y=80 g

substitute and solve for x

[tex]\begin{gathered} 80=300(0.5^)^{(\frac{x}{11})} \\ \frac{80}{300}=(0.5)^{(\frac{x}{11})} \\ apply\text{ log on both sides} \end{gathered}[/tex][tex]log\frac{80}{300}=\frac{x}{11}log(0.5)^[/tex]

x=21.0 min

the answer is 21.0 minutes