Question

A radioactive substance decays at a rate of 6% each year. If the initial amount of the substance was 600 grams, create an exponential function to model the decay of the substance. Which of the following options correctly models the decay?

Option #1: f(x)=600(0.06)x


Option #2: f(x)=600(1.06)x


Option #3: f(x)=600(0.6)x


Option #4: f(x)=600(0.94)x

Answer 1

To model the decay of a radioactive substance, we can use the exponential decay formula:

\[ f(t) = f_0 \cdot e^{-kt} \]

where:
- \( f(t) \) is the amount of substance at time \( t \),
- \( f_0 \) is the initial amount of the substance,
- \( k \) is the decay constant, and
- \( t \) is time in years.

In this case, the initial amount \( f_0 \) is 600 grams and the substance decays at a rate of 6% each year. This means that after one year, 94% of the substance remains (100% - 6% = 94%).

Thus, the decay can be modeled with the following formula:

\[ f(t) = 600 \cdot (0.94)^t \]

Now, let's check the given options to see which one matches this model.

- **Option #1:** \( f(x)=600(0.06)x \) - This represents growth and does not model decay.

- **Option #2:** \( f(x)=600(1.06)x \) - This also represents growth (an increase of 6%).

- **Option #3:** \( f(x)=600(0.6)x \) - This suggests a constant decay to 60% of the original amount, which is not correct.

- **Option #4:** \( f(x)=600(0.94)x \) - This is the correct model of decay, as it indicates that 94% of the substance remains each year.

Therefore, the correct option that models the decay of the substance is:

**Option #4: f(x)=600(0.94)x**