How long will it take a sample of radioactive substance to decay to half of its original amount, if it decays according to the functi $\mathrm { A } ( \mathrm { t } ) = 250 \mathrm { e } ^ { - .223 \mathrm { t } }$ , where t is the time in years? Round your answer to the nearest
Hundredth year.
A) 27.87 yr
B) 55.75 yr
C) 3.11 yr
D) 24.76 yr

Question

Quizplus · Accepted Answer

To find the half-life, we set $A(t) = \frac{250}{2} = 125$ and solve for $t$. Substituting into the equation gives $125 = 250e^{-0.223t}$. Dividing both sides by 250 gives $0.5 = e^{-0.223t}$. Taking the natural logarithm of both sides gives $\ln(0.5) = -0.223t$. Solving for $t$ gives $t = \frac{\ln(0.5)}{-0.223} \approx 3.11$ years.

How Long Will It Take a Sample of Radioactive Substance A(t)=250e−.223t\mathrm { A } ( \mathrm { t } ) = 250 \mathrm { e } ^ { - .223 \mathrm { t } }A(t)=250e−.223t

How Long Will It Take a Sample of Radioactive Substance $\mathrm { A } ( \mathrm { t } ) = 250 \mathrm { e } ^ { - .223 \mathrm { t } }$