To the nearest whole percentage, what is the constant percent rate of change of the exponential function y=650(0.75)0.6t

?(1 point)
The constant percent rate of change is about
%
.

1 answer

To find the constant percent rate of change of the exponential function \( y = 650(0.75)^{0.6t} \), we can rewrite it in the standard exponential form \( y = A e^{kt} \).

First, note that \( 0.75 \) can be expressed in terms of \( e \) as follows:

\[ 0.75 = e^{\ln(0.75)} \]

Thus, we can rewrite the function as:

\[ y = 650 \cdot e^{0.6 \ln(0.75) t} \]

This means we have \( k = 0.6 \ln(0.75) \).

Now we can calculate \( \ln(0.75) \):

\[ \ln(0.75) \approx -0.28768 \]

Then:

\[ k = 0.6 \cdot (-0.28768) \approx -0.172608 \]

The constant percent rate of change can be found using the formula \( r = e^{k} - 1 \). First, we need to find \( e^{-0.172608} \):

\[ e^{-0.172608} \approx 0.8403 \]

Now calculate \( r \):

\[ r = 0.8403 - 1 \approx -0.1597 \]

To convert this to a percentage, we multiply by 100:

\[ \text{Rate of change} \approx -15.97% \]

Rounding to the nearest whole percentage, the constant percent rate of change is approximately:

\[ \boxed{-16%} \]