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?

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.