Question

The half-life of a radioactive substance is one day, meaning that every day half of the substance hasdecayed. Suppose you have 100 grams of this substance.14. Construct an exponential model for the amount of the substance remaining on a given day.15. How many grams of the substance would be left after a week? (round to the hundredths place)

Answer 1

Let A₀ be the initial mass of the sample of the radioactive substance.

Since half of the substance decays each day, then, the next day the amount of radioactive substance left is:

$A_0\cdot\frac{1}{2}$

After t days, the total amount would have decayed by 1/2, t times. Then, the amount A of the radioactive substance left after t days is:

$A=A_0\cdot(\frac{1}{2})^t$

To find how many grams of the substance would be left after one week, replace A₀=100 and t=7:

$A=100\cdot(\frac{1}{2})^7=0.78125\ldots$

Therefore, the exponential model that tells the amount of substance remaining on a given day, is:

$A=100\cdot(\frac{1}{2})^t$

And the amount of grams left after a week is 0.78.