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Chapter 4: Problem 143
Without using a calculator, find the exact value of $$\frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001}$$
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Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A=4.3 e^{0.01 t}\) describes New Zealand's population, \(A,\) in millions, \(t\) years after 2010 . a. What is New Zealand's growth rate? b. How long will it take New Zealand to double its population?
Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, \(f(t),\) after \(t\) months was modeled by the function $$f(t)=88-15 \ln (t+1), \quad 0 \leq t \leq 12$$ a. What was the average score on the original exam? b. What was the average score after 2 months? 4 months? 6 months? 8 months? 10 months? one year? c. Sketch the graph of \(f\) (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students.
The bar graph indicates that the percentage of fi rst-year college students expressing antifeminist views declined after 1970. CAN'T COPY THE GRAPH The function $$f(x)=-4.82 \ln x+32.5$$ models the percentage of first-year college women, \(f(x)\) expressing antifeminist views (by agreeing with the statement) \(x\) years after 1969 a. Use the function to find the percentage of first-year college women expressing antifeminist views in 2008 . Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph? By how much? b. Use the function to project the percentage of first-year college women who will express antifeminist views in \(2015 .\) Round to one decimal place.
Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 . \quad\) Use \(\quad\) a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.
Check each proposed solution by direct substitution or with a graphing utility. $$(\ln x)^{2}=\ln x^{2}$$
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