How long will it take a sample of radioactive substance to decay to half of its original amount, if it decays according to the function $\mathrm{A}(\mathrm{t})=250 \mathrm{e}^{-0.235 \mathrm{t}}$, where $\mathrm{t}$ is the time in years? Round your answer to the nearest hundredth year.
A) $2.95 \mathrm{yr}$
B) $23.50 \mathrm{yr}$
C) $58.75 \mathrm{yr}$
D) $26.45 \mathrm{yr}$

Question

Quizplus · Accepted Answer

To find the half-life, set $A(t) = \frac{1}{2} \times 250 = 125$ and solve for $t$. Using the given function, $125 = 250e^{-0.235t}$. Dividing both sides by 250 gives $0.5 = e^{-0.235t}$. Taking the natural logarithm of both sides gives $\ln(0.5) = -0.235t$. Solving for $t$ gives $t = \frac{\ln(0.5)}{-0.235} \approx 2.95$ years.

How Long Will It Take a Sample of Radioactive Substance A(t)=250e−0.235t\mathrm{A}(\mathrm{t})=250 \mathrm{e}^{-0.235 \mathrm{t}}A(t)=250e−0.235t

How Long Will It Take a Sample of Radioactive Substance $\mathrm{A}(\mathrm{t})=250 \mathrm{e}^{-0.235 \mathrm{t}}$