MAT-111A - College Algebra Homework Problems 5 (20 Questions) The following questions were adapted from Blitzer, R. (2018). College Algebra (7th ed). NOTE: Be sure to show your work for the solution to each question. Partial credit may be given, even if the final answer is not correct, as long at the proper concepts are being depicted. Scoring: 1 point each 1. Graph the function 1 h ( x )=( ) 2 x by making a table of coordinates, then use the coordinate template below to draw the graph of h(x) . 5 4 3 2 1 -5 -4 -3 -2 -1-1 1 2 3 4 5 x -2 -3 -4 -5 2. Graph functions f ( x )= 1 2 x () and 1 x1 g ( x ) =( ) +1 2 in the same rectangular coordinate system. Graph and give equations of all asymptotes. Use the coordinate template below to draw the graph of f ( x )g(x ) . 5 4 3 2 1 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 1 2 3 4 5 x nt 3. Note: Use the compound interest formulas r A=P(1+ ) n and A=P e n to calculate the following. Round answers to the nearest cent. Find the accumulated value of an investment of $10,000 for 5 years at an interest rate of 5.5% if the money is a. compounded semiannually b. compounded quarterly c. compounded monthly d. compounded continuously 4. India is currently one of the world's fastest-growing countries. By 2040, the population of India will be larger than the population of China; by 2050, nearly one-third of the world's population will live in these two countries alone. The exponential function f ( x)=574 (1.026)x models the population of India, f(x), in millions, x years after 1974. a. Substitute 0 for x and, without using a calculator, find India's population in 1974. b. Substitute 27 for x and use your calculator to find India's population, to the nearest million, in the year 2001 as modeled by this function. c. Find India's population, to the nearest million, in the year 2028 as predicted by this function. d. Find India's population, to the nearest million, in the year 2055 as predicted by this function. e. What appears to be happening to India's population every 27 years? 5. Write the equation 5=log b 32 in its equivalent exponential form. 6. The figure below shows the graph of f(x) = log x. Use transformations of this graph to graph the function g ( x ) =log(x2) . Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. Use the coordinate template below to draw the graph of g (x ) . y 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 x -2 -4 -6 -8 -10 7. The percentage of adult height attained by a girl who is x years old can be modeled by f(x) = 62 + 35 log(x - 4), where x represents the girl's age (from 5 to 15) and f(x) represents the percentage of her adult height. Use the function to determine the approximate percentage of her adult height a girl has attained at age 13? Round answers to the nearest tenth of a percent. 8. 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), 0 ... t ... 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? 9. Use the properties of logarithms to expand the logarithmic expression log 9 ( 9x ) as much as possible. 10. Use the properties of logarithms to condense the logarithmic expression log ( 2 x +5 ) log x . Write the expression as a single logarithm whose coefficient is 1. 11. The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human ear. The formula I0 log I log D=10 describes the loudness level of a sound, D, in decibels, where I is the intensity of the sound, in watts per meter2, and I0 is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound? 12. The formula t= 1 [ ln Aln( AN ) ] c describes the time, t, in weeks, that it takes to achieve mastery of a portion of a task, where A is the maximum learning possible, N is the portion of the learning that is to be achieved, and c is a constant used to measure an individual's learning style. a. Express the formula so that the expression in brackets is written as a single logarithm. b. The formula is also used to determine how long it will take chimpanzees and apes to master a task. For example, a typical chimpanzee learning sign language can master a maximum of 65 signs. Use the form of the formula from part (a) to answer this question: How many weeks will it take a chimpanzee to master 30 signs if c for that chimp is 0.03? 13. Solve the exponential equation 9 e x =107 . Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 14. Solve the logarithmic equation ln x+3=1 . Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 15. The formula A=37.3 e 0.0095 t models the population of California, A, in millions, t years after 2010. a. What was the population of California in 2010? b. When will the population of California reach 40 million? 16. The function f ( x )=20( 0.975) x models the percentage of surface sunlight, f ( x) , that reaches a depth of x feet beneath the surface of the ocean. The figure below shows the graph of this function. Use the function to determine at what depth, to the nearest foot, there is 1% of surface sunlight. How is this shown on the graph of f? 17. The exponential models below describe the population of the indicated country, A, in millions, t years after 2010. Russia - A=141.9e-0.005t Japan - A=127.3e-0.006t Iraq - A=31.5e0.019t India - A=1173.1e0.008t a. What was the population of Japan in 2010? b. What was the population of Iraq in 2010? c. Which country has the greatest growth rate? By what percentage is the population of that country increasing each year? d. Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year? e. When will India's population be 1,377 million? f. When will India's population be 1,491 million? 18. An artifact originally had 16 grams of carbon-14 present. The decay model A=16 e0.000121 t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5,715 years? Document your calculations. kt A= A 0 e 19. Use the exponential decay model, . The half-life of thorium-229 is 7340 years. How long will it take for a sample of this substance to decay to 20% of its original amount? Round answers to one decimal place. 20. Using the data in the table below. a. Create a scatter plot for the data show below. x 1 2 3 -1 -2 -3 b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data