To model the decay of a radioactive substance, we can use the exponential decay formula:
\[ f(t) = A \cdot e^{kt} \]
or, using a base of 1 minus the decay rate:
\[ f(t) = A \cdot (1 - r)^t \]
where:
- \( A \) is the initial amount (600 grams in this case),
- \( r \) is the decay rate (6% or 0.06),
- \( t \) is the time in years.
Since the substance decays at a rate of 6%, the amount remaining after \( t \) years can be modeled as:
\[ f(t) = 600 \cdot (1 - 0.06)^t \] \[ f(t) = 600 \cdot (0.94)^t \]
Now, looking at the provided options:
- Option #1: \( f(x) = 600(0.06)^x \) (this is not correct for decay; it represents a much faster decrease)
- Option #2: \( f(x) = 600(1.06)^x \) (this represents growth, not decay)
- Option #3: \( f(x) = 600(0.6)^x \) (this does not reflect the 6% decay correctly)
- Option #4: \( f(x) = 600(0.94)^x \) (this accurately models the decay)
Thus, the correct option is:
Option #4: \( f(x) = 600(0.94)^x \)