Question

To the nearest tenth of a percentage, what is the constant percent rate of change of the exponential function y=1,350(1.15)0.1t
?(1 point)
The constant percent rate of change is about
%
.

Answer 1

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%**.