Question

in 2006, a sample of a radioactive substance had a mass of 550 milligrams. since then, the sample has decayed by 7.7% each year. let t be the number of years since 2006. let y be the mass of the substance 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 the time.

Step2: Identify values of \(a\) and \(r\)

Here, the initial mass \(a = 550\) milligrams (in 2006, when \(t = 0\)). The decay rate \(r = 7.7\% = 0.077\).

Step3: Substitute values into the formula

Substitute \(a = 550\) and \(r = 0.077\) into the exponential decay formula. So we get \(y = 550(1 - 0.077)^t\), which simplifies to \(y = 550(0.923)^t\).

Answer:

\(y = 550(0.923)^t\)