Kylie sent a chain letter to her friends, asking them to forward the letter to more friends.
The relationship between the elapsed time [tex]\(t\)[/tex], in hours, since Kylie sent the letter, and the number of people, [tex]\(P(t)\)[/tex], who receive the email is modeled by the following function: [tex]\[
P(t) = (1.09)^{6t + 25.5}
\][/tex]
Complete the following sentence about the hourly percent change in the number of people who receive the email. Round your answer to the nearest percent.
Every hour, there is a [tex]\(\square\)[/tex] [tex]\(\%\)[/tex] addition to/removal from the number of people who receive the email.
To determine the hourly percent change in the number of people receiving the email, we analyze the given exponential function:
[tex]\[ P(t) = (1.09)^{6t + 25.5} \][/tex]
Here's the step-by-step reasoning:
1. Identify the base of the exponential function: The base here is [tex]\(1.09\)[/tex].
2. Determine the percentage change per hour: The exponential growth rate is found by calculating the percentage change attributed to the base of the exponent. For an exponential function with the base [tex]\(b\)[/tex], the percentage change is given by [tex]\((b - 1) \times 100\)[/tex].
3. Calculate the hourly percentage increase: For our base [tex]\( 1.09 \)[/tex]: [tex]\[
(1.09 - 1) \times 100 = 0.09 \times 100 = 9\%
\][/tex]
4. Interpret the result: Since the base [tex]\(1.09\)[/tex] represents a growth factor, it is indicating an addition rather than a removal.
5. Conclusion: The statement should reflect that every hour, there is an addition to the number of people receiving the email.
So, completing the sentence:
"Every hour, there is a [tex]\(9\%\)[/tex] addition to the number of people who receive the email."
