There was a sample of 900 milligrams of a radioactive substance to start a study. Since then, the sample has decayed by \( 7.4 \% \) 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 \( t \). .

To model the decay of the radioactive substance, we can use the exponential decay formula: \[ y = y_0 \cdot (1 - r)^t \] where \( y_0 \) is the initial amount, \( r \) is the decay rate, and \( t \) is the time in years. Here, \( y_0 = 900 \) mg and \( r = 0.074 \) (7.4% as a decimal). Plugging these values in gives us: \[ y = 900 \cdot (1 - 0.074)^t \] \[ y = 900 \cdot (0.926)^t \] This function now describes the relationship between the mass of the sample and the time in years. Just a fun fact: This type of decay is often used when discussing half-lives, which indicate how long it takes for half of the substance to decay. For this sample, you can calculate the half-life using the formula: \[ \text{Half-life} = \frac{\ln(0.5)}{\ln(1 - 0.074)} \] Calculating that out gives you an idea of how quickly or slowly the substance would decay to half its original mass!