Question

A scientist begins with 100 milligrams of a radioactive substance that decays exponentially. After 33 hours, 50 mg of the substance remains. How many milligrams will remain after 59 hours? (Round your answer to two decimal places.)

Answer 1

Answer:

$m\approx34.99$

Explanation:

Radioactive decay behaves according to the law of exponential decay:

$m(t)=m_0*e^{-rt}$

Where:

$r=Decay\hspace{3}rate$

$t=Time(In\hspace{3}days)$

$m_0=Initial\hspace{3}amount\hspace{3}of\hspace{3}substance$

$m(t)=Amount\hspace{3}of\hspace{3}substance\hspace{3}at\hspace{3}time\hspace{3}t$

So we need to find the decay rate r. Let's find it using the equation and the data provide, but first, we need to convert the hours in days:

$50h*\frac{1day}{24h}=\frac{25}{12}day\\  33h*\frac{1day}{24h} =\frac{11}{8}day$

Now for $t=\frac{11}{8},m=50$ then:

$m(\frac{11}{8} )=100*e^{-r(\frac{11}{8}) } =50$

Divide both sides by 100:

$e^{-r(\frac{11}{8}) } =\frac{1}{2}$

Take the natural logarithm of both sides:

$-\frac{11}{8}x=ln(\frac{1}{2} )$

Multiply both sides by -8/11:

$x=\frac{-8ln(\frac{1}{2} )}{11} \approx0.504107$

Now we have found the rate decay r, let's find how many milligrams will remain after 59 hours (25/12 day):

$m(\frac{25}{12} )=100*e^{-0.504107*\frac{25}{12}) } =100*e^{-1.050222917}=34.98597508\approx35.99$