Question

A radioactive substance decays at an annual rate of $13$ percent. If the initial amount of the substance is $325$ grams, which of the following functions $f$ models the remaining amount of the substance, in grams, $t$ years later?

• A. $f\left(t\right)=325(0.87{)}^t$
• B. $f\left(t\right)=325(0.13{)}^t$
• C. $f\left(t\right)=0.87(325{)}^t$
• D. $f\left(t\right)=0.13(325{)}^t$

Answer 1

The correct option is A $f\left(t\right)=325(0.87{)}^t$

Each year, the amount of the radioactive substance is reduced by $13$ percent from the prior year’s amount; that is, each year, $87$ percent of the previous year’s amount remains.

Since the initial amount of the radioactive substance was $325$ grams, after $1$ year, $325\times 0.87$ grams remains; after $2$ years $325\times 0.87\times 0.87=325\times (0.87{)}^2$ grams remains; and after $t$ years, $325(0.87{)}^t$ grams remains.

Therefore, the function$f\left(t\right)=325(0.87{)}^t$ models the remaining amount of the substance, in grams, after $t$ years.