Exam
exam
Multiple Choice
Identify the letter of the choice that best completes the statement or answers the question.
____ 1. Determine the domain and range of the function [pic].
Select the correct answer.
|a. |Domain: [pic]; Range: [pic] |
|b. |Domain: [pic]; Range: [pic] |
|c. |Domain: [pic]; Range: [pic] |
|d. |Domain: [pic]; Range: [pic] |
|e. |Domain: [pic]; Range: [pic] |
____ 2. State the domain of the function [pic].
Select the correct answer.
|a. |[pic] |
|b. |(–4, 4) |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 3. State the range of the function
[pic].
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |(–7, 7) |
____ 4. State the asymptote of the function [pic].
Select the correct answer.
|a. |y = 3 |
|b. |y = 8 |
|c. |y = –8 |
|d. |y = –3 |
|e. |y = 0 |
____ 5. State the asymptote of the function [pic].
Select the correct answer.
|a. |y = 4 |
|b. |y = –8 |
|c. |y = –4 |
|d. |y = 0 |
|e. |y = 8 |
____ 6. State the asymptote of the function [pic].
Select the correct answer.
|a. |y = –4 |
|b. |y = 5 |
|c. |y = 0 |
|d. |y = 4 |
|e. |y = –5 |
____ 7. Determine the domain and range of the function [pic].
Select the correct answer.
|a. |Domain: [pic]; Range: [pic] |
|b. |Domain: [pic]; Range: [pic] |
|c. |Domain: [pic]; Range: [pic] |
|d. |Domain: [pic]; Range: [pic] |
|e. |Domain: [pic]; Range: [pic] |
____ 8. State the range of the function [pic].
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 9. The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. Find the present value of $10,000 if interest is paid at a rate of 7% per year, compounded semiannually, for 2 years.
Select the correct answer.
|a. |$11,475 |
|b. |$8,734 |
|c. |$8,257 |
|d. |$11,449 |
|e. |$8,714 |
____ 10. What is the asymptote of the function [pic]?
Select the correct answer.
|a. |y = 2 |
|b. |y < 2 |
|c. |x > 5 |
|d. |y = 5 |
|e. |x = 5 |
[pic]
____ 11. A 50–gallon barrel is filled completely with pure water (see the figure below). Salt water with a concentration of 0.5 lb/gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time t is given by
[pic]
where t is measured in minutes and Q(t) is measured in pounds. How much salt is in the barrel after 5 min?
|a. |5.4 pounds |
|b. |7.5 pounds |
|c. |6.5 pounds |
|d. |5.5 pounds |
|e. |7.6 pounds |
____ 12. The rat population in New York City is given by the function
[pic]
where t is measured in years since 1990 and n(t) is measured in millions. What is the rat population in 1997?
Select the correct answer.
|a. |277 million |
|b. |197 million |
|c. |18 million |
|d. |109 million |
|e. |234 million |
____ 13. A radioactive substance decays in such a way that the amount of mass remaining after t days is given by
[pic]
where m(t) is measured in kilograms. How much of the mass remains after 45 days?
Select the correct answer.
|a. |5.94 kg |
|b. |11.53 kg |
|c. |20.65 kg |
|d. |5.78 kg |
|e. |5.86 kg |
____ 14. Find the local maximum value of the function
[pic].
Select the correct answer.
|a. |8.72 |
|b. |1 |
|c. |6 |
|d. |16.31 |
|e. |2.21 |
____ 15. If $1,000 is invested at an interest rate of 6% per year, compounded monthly, find the amount of the investment at the end of 4 years.
Select the correct answer.
|a. |$1,267 |
|b. |$1,062 |
|c. |$1,262 |
|d. |$1,272 |
|e. |$1,270 |
____ 16. The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model
[pic]
where t is measured in years. What size does the population approach as time goes on?
Select the correct answer.
|a. |13,500 |
|b. |2,205 |
|c. |9,000 |
|d. |3,150 |
|e. |4,500 |
____ 17. The fox population in a certain region has a relative growth rate of 8% per year. It is estimated that the population in 1993 was 18,000. Find a function n(t) that models the population t years after 1993.
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 18. Find the value of x at which the local minimum occurs for the function
[pic].
Select the correct answer.
|a. |4.56 |
|b. |0.28 |
|c. |0.51 |
|d. |–0.39 |
|e. |–0.51 |
____ 19. Assume that the rabbit population behaves according to the logistic growth model
[pic]
where n0 is the initial rabbit population. If the initial population is 50 rabbits, what will the population be in 14 years?
|a. |9,935 rabbits |
|b. |1,078 rabbits |
|c. |1,017 rabbits |
|d. |1,656 rabbits |
|e. |1,067 rabbits |
____ 20. Radioactive iodine is used by doctors as a tracer in diagnosing certain thyroid gland disorders. This type of iodine decays in such a way that the mass remaining after t days is given by the function
[pic]
where m(t) is measured in grams. Find the mass at time t = 30.
Select the correct answer.
|a. |0 g |
|b. |8 g |
|c. |0.77 g |
|d. |0.69 g |
|e. |78 g |
____ 21. A sum of $7,000 is invested at an interest rate of 10% per year, compounded semiannually. After how many years will this investment amount to $12,571?
|a. |9 years |
|b. |8 years |
|c. |7 years |
|d. |5 years |
|e. |6 years |
____ 22. The relative growth rate for a certain bacteria population is 90% per hour. A small culture is formed and 5 hours later a count shows approximately 12,000 bacteria in the culture. Find the initial number of bacteria in the culture.
Select the correct answer.
|a. |1,080,206 |
|b. |0 |
|c. |12,009 |
|d. |133 |
|e. |1,265 |
[pic]
____ 23. A sky diver jumps from a reasonable height above the ground (see the figure below). The air resistance she experiences is proportional to her velocity and the constant of proportionality is 0.7. It can be shown that the downward velocity of the sky diver at time t is given by
[pic]
where t is measured in seconds and v(t) is measured in feet per second. Find the velocity after 10 s.
Select the correct answer.
|a. |169 feet per second |
|b. |80 feet per second |
|c. |59 feet per second |
|d. |0 feet per second |
|e. |70 feet per second |
____ 24. Under ideal conditions, a certain type of bacteria has a relative growth rate of 290% per hour. A number of these bacteria are introduced accidentally into a food product. 2 hours after contamination, a bacterium count shows that there are about 20,000 bacteria in the food. Estimate the number of bacteria in the food 3 hours after contamination.
Select the correct answer.
|a. |363,483 |
|b. |1,100 |
|c. |120,058,244 |
|d. |20,000 |
|e. |none of these |
____ 25. If $1,000 is invested at an interest rate of 10% per year, compounded semiannually, find the value of the investment after 10 years.
Select the correct answer.
|a. |$1,000 |
|b. |$2,753 |
|c. |$1,629 |
|d. |$377 |
|e. |$2,653 |
____ 26. Use the definition of the logarithmic function to find x:
[pic].
Select the correct answer.
|a. |x = 343 |
|b. |x = 2,187 |
|c. |x = 7 |
|d. |x = 21 |
|e. |none of these |
____ 27. Express the equation in exponential form.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |none of these |
____ 28. Evaluate the expression:
[pic].
Select the correct answer.
|a. |7 |
|b. |7e |
|c. |[pic] |
|d. |[pic] |
|e. |none of these |
____ 29. Express the equation in logarithmic form.
[pic]
Select the correct answer.
|a. |log 6 n = z |
|b. |log 6 z = n |
|c. |log n z = 6 |
|d. |log z 6 = n |
|e. |none of these |
____ 30. Find the domain of the function
[pic].
Select the correct answer
|a. |[2, 8) |
|b. |[–2, 8] |
|c. |[–8, –2) |
|d. |[2, 8] |
|e. |none of these |
____ 31. Use the definition of the logarithmic function to find x.
[pic]
Select the correct answer.
|a. |x = 4 |
|b. |x = 2 |
|c. |x = 16 |
|d. |x = 6 |
|e. |none of these |
____ 32. Find the domain of the function
[pic].
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |none of these |
____ 33. Evaluate the expression log 6 216.
Select the correct answer.
|a. |3 |
|b. |6 |
|c. |216 |
|d. |36 |
|e. |none of these |
____ 34. Evaluate the expression.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |1 |
|e. |none of these |
____ 35. Find the domain of the function:
[pic].
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |none of these |
____ 36. Express the equation in logarithmic form:
[pic].
Select the correct answer.
|a. |x = –6 + ln 0.3 |
|b. |x = 6 + ln 0.3 |
|c. |x = 0.3 + ln 6 |
|d. |x = 0.3 – ln 6 |
|e. |none of these |
____ 37. Express the equation ln (x + 2) = 9 in exponential form.
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |none of these |
____ 38. Express the equation in logarithmic form.
[pic]
Select the correct answer.
|a. |log 2 16 = 4 |
|b. |log 16 2 = 4 |
|c. |log 4 16 = 2 |
|d. |log 4 2 = 16 |
|e. |none of these |
____ 39. Use a calculator to evaluate the expression:
[pic].
Select the correct answer.
|a. |1.0051 |
|b. |0.4020 |
|c. |2.4121 |
|d. |0.8040 |
|e. |none of these |
____ 40. Rewrite the expression as a single logarithm.
log 3 5 + 5 log 3 2
Select the correct answer.
|a. |1 |
|b. |log 3 160 |
|c. |log 3 10 |
|d. |log 160 3 |
|e. |ln 160 |
____ 41. Simplify.
(log 3 7)(log 7 19)
Select the correct answer.
|a. |log 19 3 |
|b. |[pic] |
|c. |log 3 19 |
|d. |[pic] |
|e. |log 3 26 |
____ 42. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |log 8 x2 – 3x |
|b. |2 log 8 x – log 8 3 |
|c. |log 8 x + log 8 (x – 3) |
|d. |log 8 x – log 8 (x – 3) |
|e. |log 8 x + log 8 x – 3 |
____ 43. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 44. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 45. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |log 3 x – log 3 2 |
|c. |log 3 x + log 3 2 |
|d. |(log 3 x) (log 3 2) |
|e. |[pic] |
____ 46. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 47. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 48. Evaluate the expression.
[pic]
Select the correct answer.
|a. |80 |
|b. |12 |
|c. |81 |
|d. |5.294 |
|e. |64 |
____ 49. Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms.
log 2 7
Select the correct answer.
|a. |3.368826 |
|b. |3.817355 |
|c. |0.356207 |
|d. |2.806955 |
|e. |2.807355 |
____ 50. Evaluate the expression.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |22 |
|e. |2 |
____ 51. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 52. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 53. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 54. Rewrite the expression as a single logarithm.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 55. Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms.
log 19 2.2
|a. |0.267078 |
|b. |0.348112 |
|c. |0.267778 |
|d. |3.734430 |
|e. |0.268778 |
____ 56. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 57. Rewrite the expression below as a single logarithm.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |log 10 |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 58. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
Select the correct answer.
|a. |8 log a x – log a y + 7 log a z |
|b. |8 log a x + log a y + 7 log a z |
|c. |– 8 log a x + log a y + 7 log a z |
|d. |[pic] |
|e. |8 log a x – log a y – 7 log a z |
____ 59. Evaluate the expression.
log 3 108 – log 3 4.
Select the correct answer.
|a. |12 |
|b. |ln 108 |
|c. |3 |
|d. |4 |
|e. |log 3 104 |
____ 60. Solve the equation.
[pic]
Select the correct answer.
|a. |x = 1 |
|b. |x = 3 |
|c. |x = –9 |
|d. |x = 9 |
|e. |none of these |
____ 61. Solve the equation.
[pic]
Select the correct answer:
|a. |x = 8 |
|b. |x = 0.5808 |
|c. |x = 1.8928 |
|d. |x = 0.9031 |
|e. |x = 5 |
____ 62. Find the solution of the exponential equation, correct to four decimal places.
[pic]
Select the correct answer.
|a. |x = 2.7183 |
|b. |x = 0.8074 |
|c. |x = 0.5404 |
|d. |x = –14.0832 |
|e. |x = –0.213 |
____ 63. Solve the inequality.
[pic]
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 64. Find the solution of the exponential equation, correct to four decimal places.
[pic]
Select the correct answer.
|a. |x = 61.0572 |
|b. |x = 3 |
|c. |x = –3.006 |
|d. |x = –61.0572 |
|e. |x = 0.3333 |
____ 65. A 22–g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by
[pic]
where m( t ) is measured in grams. After how many days is there only 12 g remaining?
Select the correct answer.
|a. |7 days |
|b. |10 days |
|c. |8 days |
|d. |9 days |
|e. |6 days |
____ 66. Find the solution of the exponential equation, correct to four decimal places.
[pic]
Select the correct answer.
|a. |x = 5.1285 |
|b. |x = 1.585 |
|c. |x = 0.585 |
|d. |x = 5.1297 |
|e. |none of these |
____ 67. Solve the logarithmic equation for x.
log 3( 4 – x ) = 2
Select the correct answer.
|a. |x = –5 |
|b. |x = 9 |
|c. |x = –13 |
|d. |x = 13 |
|e. |x = –9 |
____ 68. Solve the inequality.
[pic]
Select the correct answer:
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 69. Solve the logarithmic equation for x.
log 4( x + 7 ) – log 4( x – 7 ) = 4
Select the correct answer.
|a. |x = 85.67 |
|b. |x = 6.95 |
|c. |x = 4 |
|d. |x = 7.05 |
|e. |x = 0.14 |
____ 70. Solve the equation.
[pic]
Select the correct answer.
|a. |x = 0.4551, x = 0 |
|b. |x = 2.1972, x = 0 |
|c. |x = 9, x = 1 |
|d. |x = 2.3026 |
|e. |x = –9, x = 1 |
____ 71. Find the solution of the exponential equation, correct to four decimal places.
[pic]
Select the correct answer.
|a. |x = 0.7325 |
|b. |x = 1.3652 |
|c. |x = 0.5556 |
|d. |x = 1.8 |
|e. |x = 0.5566 |
____ 72. For what value of x is the following true?
log ( x + 21 ) = log x + log 21
Select the correct answer.
|a. |x = –19.95 |
|b. |x = 1.05 |
|c. |x = 22 |
|d. |x = 10.5 |
|e. |x = 0.021 |
____ 73. Solve the logarithmic equation for x.
log x = 5
Select the correct answer.
|a. |x = 50,000 |
|b. |x = 100,000 |
|c. |x = 0.699 |
|d. |x = 10,000 |
|e. |x = 0.00001 |
____ 74. Find the solution of the exponential equation, correct to four decimal places.
[pic]
Select the correct.
|a. |x = 2.4881 |
|b. |x = –3.5693 |
|c. |x = 1.4247 |
|d. |x = 1.6077 |
|e. |x = 1.7077 |
____ 75. Solve the inequality.
log ( x – 3 ) + log ( 10 – x ) < 1
Select the correct answer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |[pic] |
____ 76. Solve the logarithmic equation for x.
log 22 + log 2x = log 24 + log 2( x – 10 )
Select the correct answer.
|a. |x = 20 |
|b. |x = 22 |
|c. |x = 16 |
|d. |x = 4.3 |
|e. |x = 40 |
____ 77. Solve the logarithmic equation for x.
ln x = 10
Select the correct answer.
|a. |x = –100 |
|b. |x = 22,026.4658 |
|c. |x = 10 |
|d. |x = 11,013.2329 |
|e. |x = 8,103.0839 |
____ 78. Solve the logarithmic equation for x.
log ( 2x + 8 ) = 3
Select the correct answer.
|a. |x = 496 |
|b. |x = 248 |
|c. |x = 124 |
|d. |x = 551.7 |
|e. |none of these |
____ 79. Find the solution of the exponential equation below, correct to four decimal places.
[pic]
Select the correct answer.
|a. |x = –5.7823 |
|b. |x = –0.6945 |
|c. |x = 0.6931 |
|d. |x = 3.1699 |
|e. |x = 5.7708 |
____ 80. The pH reading of a glass of beer is 4.7. Find the hydrogen ion concentration of the beer.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |none of these |
____ 81. The noise from a power mower was measured at 107 dB. The noise level at a rock concert was measured at 120 dB. Find the ratio of the intensity of the rock music to that of the power mower.
|a. |3.7 |
|b. |20.0 |
|c. |22.6 |
|d. |13.0 |
|e. |8.1 |
____ 82. A man invests $15,000 in an account that pays 7.5% interest per year, compounded quarterly. Find the amount after 3 years.
|a. |$18,745.75 |
|b. |$18,634.45 |
|c. |$15,859.67 |
|d. |$35,726.69 |
|e. |none of these |
____ 83. A culture starts with 8,300 bacteria. After one hour the count is 10,000. Find a function that models the number of bacteria n ( t ) after t hours.
|a. |n ( t ) = 8,300e0.25t |
|b. |n ( t ) = 10,000e0.19t |
|c. |n ( t ) = 10,000e0.22t |
|d. |n ( t ) = 8,300e0.22t |
|e. |n ( t ) = 8,300e0.19t |
____ 84. The intensity level of the sound of a subway train was measured at 90 dB. Find the intensity in W / m 2.
|a. |[pic] |
|b. |[pic] |
|c. |[pic] |
|d. |[pic] |
|e. |none of these |
____ 85. How long will it take for an investment of $700 to double in value if the interest rate is 8.5% per year, compounded continuously?
|a. |8.15 years |
|b. |12.92 years |
|c. |12.7 years |
|d. |0.08 year |
|e. |none of these |
____ 86. If one earthquake is 12 times as intense as another, how much larger is its magnitude on the Richter scale?
|a. |2.48 larger on the Richter scale |
|b. |1.78 larger on the Richter scale |
|c. |1.38 larger on the Richter scale |
|d. |1.08 larger on the Richter scale |
|e. |2.01 larger on the Richter scale |
____ 87. The frog population in a small pond grows exponentially. The current population is 70 frogs, and the relative growth rate is 20% per year. Find the projected population after 5 years.
|a. |22,026 frogs |
|b. |3 frogs |
|c. |77 frogs |
|d. |209 frogs |
|e. |190 frogs |
____ 88. A kettle full of water is brought to a boil in a room with temperature 24ºC. After 10 min the temperature of the water has decreased from 100ºC to 70ºC. Find the temperature after another 7 min.
|a. |32ºC |
|b. |70ºC |
|c. |67ºC |
|d. |8ºC |
|e. |56ºC |
____ 89. Radium–221 has a half–life of 30 s. How long will it take for 70% of a sample to decay?
|a. |36.12 s |
|b. |52.11 s |
|c. |10.70 s |
|d. |15.44 s |
|e. |18.06 s |
____ 90. A sum of $5,000 was invested for 6 years, and the interest was compounded semiannually. If this sum amounted to $7,300 in the given time, what was the interest rate?
|a. |6.41 % |
|b. |6.51 % |
|c. |6.36 % |
|d. |7.51 % |
|e. |none of these |
____ 91. The burial cloth of an Egyptian mummy is estimated to contain 51% of the carbon–14 it contained originally. How long ago was the mummy buried? (The half–life of carbon–14 is 5730 years.)
|a. |3,858 yr |
|b. |1,929 yr |
|c. |5,663 yr |
|d. |5,566 yr |
|e. |3,838 yr |
____ 92. The population of a certain city was 112,000 in 1990, and the observed relative growth rate is 5% per year. In what year will the population reach 220,000?
|a. |2,003 |
|b. |2,004 |
|c. |2,000 |
|d. |13 |
|e. |none of these |
____ 93. The half–life of cesium–137 is 30 years. Suppose we have a 66–g sample. Find a function that models the mass remaining after t years.
|a. |m ( t ) = 69e– 0.03t |
|b. |m ( t ) = 66e– 0.023t |
|c. |m ( t ) = 69e– 0.02t |
|d. |m ( t ) = 66e– 0.024t |
|e. |m ( t ) = 30e– 0.023t |
____ 94. Find the time required for an investment of $6,000 to grow to $12,000 at an interest rate of 9.5% per year, compounded quarterly.
|a. |7 years |
|b. |8 years |
|c. |30 years |
|d. |2 years |
|e. |none of these |
____ 95. If 250 mg of a radioactive element decays to 220 mg in 48 hours, find the half–life of the element (in hours).
|a. |750.98 h |
|b. |375.49 h |
|c. |541.72 h |
|d. |33.27 h |
|e. |260.27 h |
____ 96. The number of bacteria in a culture is modeled by the function
n ( t ) = 600e0.25t
where t is measured in hours. What is the initial number of bacteria?
|a. |600 bacteria |
|b. |1 bacteria |
|c. |25 bacteria |
|d. |1,200 bacteria |
|e. |0 bacteria |
____ 97. The population of California was 10,939,035 in 1960 and 23,186,007 in 1985. Assume the population grows exponentially. Find the time required for the population to double (in years).
|a. |26.77 yr |
|b. |23.07 yr |
|c. |0.92 yr |
|d. |33.28 yr |
|e. |66.56 yr |
____ 98. An infectious strain of bacteria increases in number at a relative growth rate of 200% per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 35 hours. How long (in hours) will it take for the critical level to be reached if the same person is infected with 16 bacteria?
|a. |33.61 h |
|b. |68.61 h |
|c. |36.39 h |
|d. |32.23 h |
|e. |37.77 h |
____ 99. An unknown substance has a hydrogen ion concentration of
[pic]
Find the pH.
|a. |pH = 3.3 |
|b. |pH = 7.7 |
|c. |pH = 4.0 |
|d. |pH = 9.0 |
|e. |none of these |
Matching
Determine the domain and range of the function [pic], then match each word with the corresponding interval.
|a. |Domain |b. |Range |
____ 100. [pic]
____ 101. [pic]
Determine the domain and range of the function [pic], then match each word with the corresponding interval.
|a. |Domain |b. |Range |
____ 102. [pic]
____ 103. [pic]
State the domain, range, and asymptote of the function [pic]. Match the words with the corresponding expressions.
|a. |domain |c. |asymptote |
|b. |range |
____ 104. [pic]
____ 105. [pic]
____ 106. [pic]
Short Answer
107. State the asymptote of the function [pic].
108. State the asymptote of the function [pic].
109. State the asymptote of the function [pic].
110. State the domain of the function [pic].
111. State the range of the function.
[pic]
112. If $10,000 is invested at an interest rate of 8% per year, compounded semiannually, find the value of the investment after 5 years.
113. The rat population in New York City is given by the function
[pic]
where t is measured in years since 1990 and n(t) is measured in millions. What is the rat population in 1996 (in millions)?
114. Under ideal conditions, a certain type of bacteria has a relative growth rate of 230% per hour. A number of these bacteria are introduced accidentally into a food product. 3 hours after contamination, a bacterium count shows that there are about 50,000 bacteria in the food. Estimate the number of bacteria in the food 6 hours after contamination.
115. What is the range of the function [pic]?
[pic]
116. A sky diver jumps from a reasonable height above the ground (see the figure below). The air resistance she experiences is proportional to her velocity and the constant of proportionality is 0.2. It can be shown that the downward velocity of the sky diver at time t is given by
[pic]
where t is measured in seconds and v(t) is measured in feet per second. Find the velocity after 5 s.
117. Assume that the rabbit population behaves according to the logistic growth model
[pic]
where n0 is the initial rabbit population. If the initial population is 50 rabbits, what will the population be in 8 years?
118. A sum of $10,000 is invested at an interest rate of 10% per year, compounded semiannually. Use a table or a graph to determine in how many years this investment will amount to $17,959?
119. Use your graphing calculator to solve the following problem.
Find the local minimum value of the following function and the value of x at which it occurs.
[pic]
State the answer correct to two decimal places.
120. The relative growth rate for a certain bacteria population is 80% per hour. A small culture is formed and 4 hours later a count shows approximately 14,500 bacteria in the culture. Find the initial number of bacteria in the culture.
121. A radioactive substance decays in such a way that the amount of mass remaining after t days is given by
[pic]
where m(t) is measured in kilograms. How much of the mass remains after 40 days?
122. The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model
[pic]
where t is measured in years. What size does the population approach as time goes on?
123. The fox population in a certain region has a relative growth rate of 10% per year. It is estimated that the population in 1998 was 13,000. Find a function n(t) that models the population t years after 1998.
124. Use your graphing calculator to solve the following problem.
Find the local maximum value of the following function and the value of x at which it occurs.
[pic]
State the answer correct to two decimal places.
125. Radioactive iodine is used by doctors as a tracer in diagnosing certain thyroid gland disorders. This type of iodine decays in such a way that the mass remaining after t days is given by the function
[pic]
where m(t) is measured in grams. Find the mass at time t = 0.
[pic]
126. A 50–gallon barrel is filled completely with pure water (see the figure below). Salt water with a concentration of 0.3 lb/gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time t is given by
[pic]
where t is measured in minutes and Q(t) is measured in pounds. How much salt is in the barrel after 5 min?
127. The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. Find the present value of $1,000 if interest is paid at a rate of 6% per year, compounded semiannually, for 10 years.
128. Use the definition of the logarithmic function to find x.
[pic]
129. Find the domain of the function
[pic].
130. Find the domain of the function
[pic].
Please enter your answer in interval notation.
131. Express the equation in exponential form.
ln (x + 4) = 5
132. Evaluate the expression.
[pic]
133. Express the equation in logarithmic form.
[pic]
134. Express the equation in logarithmic form.
[pic]
135. Use the definition of the logarithmic function to find x.
[pic]
136. Express the equation in exponential form.
[pic]
137. Express the equation in logarithmic form.
[pic]
138. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
139. Rewrite the expression below as a single logarithm.
[pic]
140. Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms.
log 15 2.1
141. Evaluate the expression.
[pic]
142. Rewrite the expression as a single logarithm.
log 7 2 + 2 log 7 2
143. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
144. Use the Change of Base Formula and a calculator to evaluate the following logarithm, correct to six decimal places. Use either natural or common logarithms.
log 4 13
145. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
146. Simplify.
(log 3 5)(log 5 17)
147. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
148. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
149. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
150. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
151. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
152. Evaluate the expression.
[pic]
153. Evaluate the expression.
log 2 24 – log 2 3.
154. Use the Laws of Logarithms to rewrite the expression
[pic]
in a form with no logarithm of a product, quotient, or power.
155. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
156. Rewrite the expression as a single logarithm.
[pic]
157. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient, or power.
[pic]
158. A 13–g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by
[pic]
where m( t ) is measured in grams. After how many days is there only 10 g remaining?
159. Solve the equation.
[pic]
160. Solve the logarithmic equation for x.
log x = –1
161. Find the solution of the exponential equation , correct to four decimal places.
[pic]
162. Find the solution of the exponential equation, correct to four decimal places.
[pic]
163. Solve the equation.
[pic]
164. Find the solution of the exponential equation, correct to four decimal places.
[pic]
165. Solve the logarithmic equation for x.
ln x = 13
166. Solve the inequality.
[pic]
167. Solve the inequality.
[pic]
168. Solve the inequality.
log ( x – 3 ) + log ( 10 – x ) < 1
169. Find the solution of the exponential equation, correct to four decimal places.
[pic]
170. Find the solution of the exponential equation, correct to four decimal places.
[pic]
171. Solve the equation.
[pic]
172. Solve the logarithmic equation for x.
log 3( 7 – x ) = 7
173. Find the solution of the exponential equation below, correct to four decimal places.
[pic]
174. For what value of x is the following true?
log ( x + 2 ) = log x + log 2
175. Solve the logarithmic equation for x.
log 27 + log 2x = log 28 + log 2( x – 1 )
176. Solve the logarithmic equation for x.
log ( 3x + 8 ) = 1
177. Solve the logarithmic equation for x.
log 2( x + 7 ) – log 2( x – 7 ) = 3
178. The noise from a power mower was measured at 107 dB. The noise level at a rock concert was measured at 121 dB. Find the ratio of the intensity of the rock music to that of the power mower.
179. The population of a certain city was 113,000 in 1994, and the observed relative growth rate is 4% per year. In what year will the population reach 206,000?
180. An unknown substance has a hydrogen ion concentration of
[pic].
Find the pH.
181. The pH reading of a glass of beer is 4.7. Find the hydrogen ion concentration of the beer.
182. The frog population in a small pond grows exponentially. The current population is 71 frogs, and the relative growth rate is 16% per year. Find the projected population after 6 years.
183. How long will it take for an investment of $700 to double in value if the interest rate is 9% per year, compounded continuously? Give the answer in years.
184. The population of California was 10,917,994 in 1960 and 23,666,274 in 1980. Assume the population grows exponentially. Find the time required for the population to double (in years).
185. If one earthquake is 19 times as intense as another, how much larger is its magnitude on the Richter scale?
186. A culture starts with 8,200 bacteria. After one hour the count is 10,000. Find a function that models the number of bacteria n( t ) after t hours.
187. The number of bacteria in a culture is modeled by the function
n ( t ) = 900e0.75t
where t is measured in hours. What is the initial number of bacteria?
188. Find the time (in years) required for an investment of $6,000 to grow to $10,000 at an interest rate of 9.5% per year, compounded quarterly.
189. An infectious strain of bacteria increases in number at a relative growth rate of 200% per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 40 hours. How long (in hours) will it take for the critical level to be reached if the same person is infected with 13 bacteria?
190. Radium–221 has a half–life of 30 s. How long will it take for 80% of a sample to decay?
191. The burial cloth of an Egyptian mummy is estimated to contain 51% of the carbon–14 it contained originally. How long ago was the mummy buried? (The half–life of carbon–14 is 5730 years.)
192. A sum of $9,000 was invested for 8 years, and the interest was compounded semiannually. If this sum amounted to $11,800 in the given time, what was the interest rate?
193. The intensity level of the sound of a subway train was measured at 95 dB. Find the intensity in W / m 2.
194. A man invests $20,000 in an account that pays 8% interest per year, compounded quarterly. Find the amount after 7 years.
195. A kettle full of water is brought to a boil in a room with temperature 23ºC. After 10 min the temperature of the water has decreased from 100ºC to 75ºC. Find the temperature after another 9 min.
196. The half–life of cesium–137 is 30 years. Suppose we have a 42–g sample. Find a function that models the mass remaining after t years.
197. If 270 mg of a radioactive element decays to 220 mg in 46 hours, find the half–life of the element (in hours).
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