To find the constant percent rate of change of the exponential function \( y = 1,350(1.15)^{0.1t} \), we can identify the growth factor from the standard form of an exponential function, which is generally expressed as \( y = A(1 + r)^t \), where \( r \) is the constant percentage rate of change.
In your function:
\[ y = 1,350(1.15)^{0.1t} \]
we can rewrite it in a more recognizable form for finding the growth rate per time unit:
\[ y = 1,350 e^{\ln(1.15) \cdot 0.1t} \]
Here, \( \ln(1.15) \) is the natural logarithm of the growth factor \( 1.15 \).
To determine the constant percent rate of change, we can compute the effective growth rate per unit of \( t \) which is \( \ln(1.15) \cdot 0.1 \) and then convert it to a percentage.
Calculating \( \ln(1.15) \):
\[ \ln(1.15) \approx 0.13976 \]
Now, multiply this by \( 0.1 \):
\[ \text{Growth Rate} = 0.1 \cdot \ln(1.15) \approx 0.1 \cdot 0.13976 \approx 0.013976 \]
To convert this to a percentage, multiply by 100:
\[ \text{Percent Rate of Change} \approx 0.013976 \cdot 100 \approx 1.3976% \]
Rounding to the nearest tenth of a percentage:
\[ \text{Percent Rate of Change} \approx 1.4% \]
Thus, the constant percent rate of change is about 1.4%.
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