Question
What is the exponential decay rate (percent rate of change) of the equation below? \( f(x)=(.85)^{x} \) \( 85 \% \) \( 15 \% \) \( 25 \% \) \( 0.85 \% \)
Ask by Dickson Hart. in the United States
Mar 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The exponential decay rate is 15%.
Solution
1. The function is given by \( f(x) = (0.85)^x \).
2. In an exponential decay function of the form \( f(x) = a^x \), the base \( a \) represents the factor by which the function changes for each unit increase in \( x \).
3. Here, \( a = 0.85 \), which means the function retains \( 85\% \) of its value per unit increase in \( x \).
4. The decay rate is the percentage by which the function decreases each time step, calculated as:
\[
\text{Decay Rate} = (1 - a) \times 100\%
\]
5. Substituting \( a = 0.85 \) into the formula:
\[
\text{Decay Rate} = (1 - 0.85) \times 100\% = 0.15 \times 100\% = 15\%
\]
6. Therefore, the exponential decay rate of the equation \( f(x)=(0.85)^x \) is \( 15\% \).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Extra Insights
The function \( f(x) = (0.85)^{x} \) represents exponential decay, where the base \( 0.85 \) indicates that the quantity decreases to \( 85\% \) of its value for each unit increase in \( x \). Specifically, the decay rate can be determined by subtracting the base from \( 1\): \( 1 - 0.85 = 0.15 \) or \( 15\% \). In practical terms, this means if you start with a value of \( 100 \), after one unit of \( x \), you'd have \( 85 \), after two units \( 72.25 \), and so on. So, if you’re tracking something that decays over time—like a radioactive substance or investment value—this equation is your best friend!
