To determine whether the given exponential function represents growth or decay, and the percentage rate of that change, we need to analyze the function in its standard exponential form.
Given:
[tex]\[ y = 1800e^{\ln(0.87)x} \][/tex]
### Step-by-Step Solution:
1. Identify the Exponential Base:
- The exponent part of the function is [tex]\( e^{\ln(0.87)x} \)[/tex].
2. Simplify the Exponent:
- Recall the property of logarithms: [tex]\( e^{\ln a} = a \)[/tex]. Here, we have [tex]\( \ln(0.87) \)[/tex], which can be rewritten as:
[tex]\[
y = 1800 \cdot (e^{\ln(0.87)})^x
\][/tex]
[tex]\[
y = 1800 \cdot 0.87^x
\][/tex]
3. Determine Growth or Decay:
- Notice the base 0.87. Since 0.87 is less than 1, the exponential function [tex]\( 0.87^x \)[/tex] represents exponential decay.
4. Calculate the Percentage Rate of Decrease:
- The decay rate can be found by recognizing that the base of 0.87 implies a decrease from 1.
- The percentage decrease per unit [tex]\( x \)[/tex] is calculated as:
[tex]\[
1 - 0.87 = 0.13 \text{ (or 13%)}
\][/tex]
### Conclusion:
The given function represents exponential decay, and the percentage rate of decrease per unit of [tex]\( x \)[/tex] is [tex]\( 13.0\% \)[/tex].
So the final answer is:
Decay [tex]\( \, \, 13.0\% \)[/tex] decrease
