Sure! Let's find the base for an exponential function that models a 15% decay.
When dealing with exponential decay, the general form of the exponential function is:
[tex]\[
y = \left( b \right)^{\frac{t}{12}}
\][/tex]
where \( b \) is the base we need to find, and \( t \) represents time.
1. Understanding the decay rate: A 15% decay implies that each unit time period, the quantity reduces by 15%. This means after one time period, 85% (which is \(100\% - 15\% = 85\%\) or \(0.85\) as a decimal) of the original amount remains.
2. Relating decay rate to base: The base \( b \) corresponds to the fraction of the quantity that remains after each unit time period. For a 15% decay per unit time period, the remaining portion is \( 0.85 \).
3. Exponent time frame adjustment: Since we are considering decay over a period of \( \frac{t}{12} \), and we want to find the equivalent base that describes decay each month, we need the base \( b \) such that:
[tex]\[
b = 0.85
\][/tex]
So, the base that models a 15% decay in the exponential function \( y = \left( b \right)^{\frac{t}{12}} \) is:
[tex]\[
b = 0.85
\][/tex]
Therefore, the function can be written as:
[tex]\[
y = (0.85)^{\frac{t}{12}}
\][/tex]
So, the value that should be placed in the blank to model a 15% decay is [tex]\( \boxed{0.85} \)[/tex].
