Question

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?

Answer 1

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 $A(t)=a\cdot b^t$, 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.

$100=200\cdot b^{24}$

Let us solve for b.

$\frac{100}{200}=\frac{200\cdot b^{24}}{200}$

$0.5=b^{24}$

$b=0.5^{\frac{1}{24}}$

So our required function is $A(t)=100\times 0.5^{\frac{1}{24}*t}$.

Substitute $t=37$ in above equation:

$A(37)=100\times 0.5^{\frac{1}{24}*37}$

$A(37)=100\times 0.5^{\frac{37}{24}}$

$A(37)=100\times 0.3434884118645223$

$A(37)=34.34884118645223$

$A(37)\approx 34.35$

Therefore, 34.35 milligrams of substance will remain after 37 hours.