To solve this problem, we'll use the exponential decay formula, which is of the form:
[tex]\[ f(t) = a \cdot b^t \][/tex]
Here's what each constant represents in our problem:
1. [tex]\( a \)[/tex]: This is the initial quantity of the substance.
2. [tex]\( b \)[/tex]: This is the base of the exponential function, and it represents the decay factor per time period [tex]\( t \)[/tex].
Given the information:
- The initial quantity of the chemical, [tex]\( a \)[/tex], is 200 milligrams.
- The decay rate is 50% over a 17-year period.
The decay rate of 50% means that after every 17 years, the quantity of the chemical is reduced by half. This means the decay factor [tex]\( b \)[/tex] for a 17-year period is:
[tex]\[ b = 1 - 0.50 = 0.50 \][/tex]
Thus, the base [tex]\( b \)[/tex] remains constant at 0.5.
To write the exponential decay formula, plug in the values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ f(t) = 200 \cdot (0.5)^t \][/tex]
Rounding these values to three decimal places, we have:
[tex]\[ f(t) = 200 \cdot 0.500^t \][/tex]
Therefore, the exponential decay formula representing the situation is:
[tex]\[ f(t) = 200 \cdot 0.500^t \][/tex]
