Question

there was a sample of 450 milligrams of a radioactive substance to start a study. since then, the sample has decayed by 9.6% each year.
let t be the number of years since the start of the study. let y be the mass of the sample in milligrams.
write an exponential function showing the relationship between y and t.

Explanation:

Step1: Recall Exponential Decay Formula

The general form of an exponential decay function is \( y = a(1 - r)^t \), where \( a \) is the initial amount, \( r \) is the rate of decay (in decimal), and \( t \) is time.
Here, \( a = 450 \) (initial mass in milligrams), and the decay rate \( r = 9.6\% = 0.096 \).

Step2: Substitute Values into Formula

Substitute \( a = 450 \) and \( r = 0.096 \) into the formula. So \( 1 - r = 1 - 0.096 = 0.904 \).
Thus, the exponential function is \( y = 450(0.904)^t \).

Answer:

\( y = 450(0.904)^t \)