Question
A psychologist is studying the capacity of the human mind to process information. He has found that the percentage of information one particular participant can recall after \( t \) months have passed can be modeled by the following logarithmic function. \[ f(t)=89-23 \ln (t+1) \text {, where } 1 \leq t \leq 12 \] What percentage of information is retained after 10 months? Round your answer to the nearest whole percentage, if necessary. Answer 2 Points Prev Keyboard Shortcuts Nexpt
Ask by John Schneider. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
After 10 months, approximately 34% of the information is retained.
Solution
To find the percentage of information retained after 10 months, substitute \( t = 10 \) into the function:
\[
f(10) = 89 - 23 \ln(10 + 1) = 89 - 23 \ln(11)
\]
Calculate \( \ln(11) \):
\[
\ln(11) \approx 2.3979
\]
Multiply by 23:
\[
23 \times 2.3979 \approx 55.15
\]
Subtract from 89:
\[
89 - 55.15 \approx 33.85\%
\]
Rounded to the nearest whole percentage:
**34%**
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Simplify this solution The Deep Dive
To find the percentage of information retained after 10 months, we can plug \( t = 10 \) into the function \( f(t) \). \[ f(10) = 89 - 23 \ln(10 + 1) \\ = 89 - 23 \ln(11) \] Now, calculate \( \ln(11) \). Approximately \( \ln(11) \approx 2.3979 \). Substituting back, we get: \[ f(10) \approx 89 - 23 \times 2.3979 \\ \approx 89 - 55.1557 \\ \approx 33.8443 \] Rounding to the nearest whole percentage, the percentage of information retained after 10 months is approximately **34%**.
