Q:

A radioactive substance decays exponentially. A scientist begins (t=0) with 200 milligrams of a radioactive substance, where the variable t t represents time (in hours). After 24 hours, 100 mg of the substance remain. How many milligrams will remain at t= 37 hours?

Accepted Solution

A:
Answer:34.35 mg.Step-by-step explanation:We have been given that a radioactive substance decays exponentially. A scientist begins (t=0) with 200 milligrams of a radioactive substance, where the variable t represents time (in hours). After 24 hours, 100 mg of the substance remain.We know that an exponential decay function is in form [tex]A(t)=a\cdot b^t[/tex], where,A(t) = Final amount,a = Initial value,b = Decay rate,t = time.For our problem initial value (a) is 200, final amount is 100 and time is 24.[tex]100=200\cdot b^{24}[/tex]Let us solve for b.[tex]\frac{100}{200}=\frac{200\cdot b^{24}}{200}[/tex][tex]0.5=b^{24}[/tex][tex]b=0.5^{\frac{1}{24}}[/tex]So our required function is [tex]A(t)=100\times 0.5^{\frac{1}{24}*t}[/tex].Substitute [tex]t=37[/tex] in above equation:[tex]A(37)=100\times 0.5^{\frac{1}{24}*37}[/tex][tex]A(37)=100\times 0.5^{\frac{37}{24}}[/tex][tex]A(37)=100\times 0.3434884118645223[/tex][tex]A(37)=34.34884118645223[/tex][tex]A(37)\approx 34.35[/tex]Therefore, 34.35 milligrams of substance will remain after 37 hours.