To model the decay of a radioactive substance, we use an exponential decay function. The general form of an exponential decay function is:
\[ f(t) = A_0 e^{-kt} \]
or in a common simpler form:
\[ f(t) = A_0 (1 - r)^t \]
where:
- \( A_0 \) is the initial amount,
- \( r \) is the decay rate (expressed as a decimal),
- \( t \) is time in years.
In your case:
- The initial amount \( A_0 = 600 \) grams,
- The decay rate \( r = 6% = 0.06 \).
So, the function becomes:
\[ f(t) = 600(1 - 0.06)^t = 600(0.94)^t \]
Now, let's analyze the given options:
- Option #1: \( f(x) = 600(0.06)^x \) (incorrect, this represents 6% of the initial amount at each time).
- Option #2: \( f(x) = 600(1.06)^x \) (incorrect, this represents growth, not decay).
- Option #3: \( f(x) = 600(0.6)^x \) (incorrect, this suggests a 40% decay, not 6%).
- Option #4: \( f(x) = 600(0.94)^x \) (correct; this represents a decay of 6% per year).
Therefore, the option that correctly models the decay of the substance is Option #4.
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