The reason we use the perimeter formula is because we know enough information about the perimeter that the formula will allow us to solve for one of the unknowns. 1. f (t) = A 0 e−0.0000000087t 27. 1+6.113686306e −0.3852149008x . a a(1+ 365(1) r ) 365 = a[(1+ −a = a 1 3 8. 21. logn (103) = Evaluate 53x when x 9. inverse exponential function. Domain: (−3, ∞); Vertical asymptote: x = −3; Let f be the exponential decay function f (x) ln(7) (5) = log (5) 61. 3. g(x) = 4(3) ; y-intercept: (0, 4); Domain: all real numbers; Range: all real numbers greater than 0. 3. 31. 47, 622 fox 35. f (x) = 4 If a quantity is independent of a variable, we usually just add or subtract it according to the problem. x −2x t 4. If the population continues to grow at this rate, there will be about 25,634 seals in 2020. 1. regression analysis command from the STAT then CALC menu. 15. p 2. 5 No. value. What was her speed on Sunday? log (n) 2 ln x 6. f (x) = √2(√2) . 5 101 If the average number of minutes used each month is 420, then Company A offers a lower monthly cost of $55 compared to Company B’s monthly cost of $56.80. 31. Credit: Kevin Dooley. The screencast video solutions here are second to none. This expression represents a variable cost because it changes according to the number of miles driven. positive base b 1 (x) and g(x) appear to be the same; Conjecture: for any = −log2 (x) 2 ≠ 1, logb (x) = −log 1 (x) . 5. log3 16 . 16/38 9/27/2020 Answer Key Chapter 6 - College Algebra | OpenStax 6.5 Section Exercises Domain: (−∞, 0) ; Range: (−∞, ∞) ; Vertical asymptote: x 29. x−3 − 6 − 6(x−3) It takes Andrew 30 minutes to drive to work in the morning. 2. less than 230 years, 229.3157 to be exact 3 7. y = 3e (ln 0.5)x 6.8 Fitting Exponential Models to Data 7/38 9/27/2020 Answer Key Chapter 6 - College Algebra | OpenStax $13, 268.58 Answers will vary. Domain: (5, ∞) ; Vertical asymptote: x = When dealing with real-world applications, there are certain expressions that we can translate directly into math. 49. f (x) = 3log4 (x + 2) x−4 > 0 . Both the PDF and printed versions of these textbooks contain Then, by the change-of-base formula, Privacy 11. in general form is all real numbers. 7. f (0) ≈ 16.7; The amount initially present is about 16.7 units. It is often helpful to make a sketch and label the sides as shown below. 10 The forest represented by the function B(t) = 82(1.029) . = − 1 Dychko, an expert physics teacher, guide you through every step. Write a linear equation to express the relationship between unknown quantities. Since the functions are inverses, their graphs are mirror images about the line y = x. When f (x) = 250, x ≈ 3.6 39. Consistent units of measurement are key to obtaining a correct solution. Find the dimensions of a box given information about its side lengths. Recall that the argument of a logarithmic function must be positive, so we determine where However, we can write the length in terms of the width as [latex]L=W+3[/latex]. 59. f −1 ln(a)−ln( (x) = c note that the graph lies above the x-axis on the interval and again to the right of the vertical asymptote, that is (4, ∞) . 5/38 9/27/2020 Answer Key Chapter 6 - College Algebra | OpenStax 7. Note that when rate is given in mi/h, time must be expressed in hours. 1. As both perimeter and area use length and width as dimensions, they are often used together to solve a problem such as this one. C x+2 1 13. 2.81359 Let y x = b 2 for some non-negative real number b such that b ≠ 1. b 53. Created by … S0 ) = 2 billion, or 1 million people. Examples of formulas include the area of a rectangular region, [latex]A=LW[/latex]; the perimeter of a rectangle, [latex]P=2L+2W[/latex]; and the volume of a rectangular solid, [latex]V=LWH[/latex]. Write an equation interpreting the words as mathematical operations. bound. 15. en = w Shifting the function right or left and reflecting the function about the y-axis will affect its domain. The equation has no solution. 19. log19 y = x text, but otherwise the end of chapter problems are identical. 65. 5. x = − 7. n = For example, populations cannot 13. logb (7) or 2 orders of magnitude greater, than the mass of Earth. On Sunday evening, due to heavy traffic, it took Jennifer 4 hours to return home. 9−e 5 x ( 23. The domain is (−∞, ∞) ; the range is (0, ∞) ; the horizontal asymptote is y = 0. final answer to quickly see if you got it right. 33. f (t) = 250e(−0.00914t) ; half-life: about 76 minutes 7. since you will see how Shaun Dychko made the calculation. Course Hero is not sponsored or endorsed by any college or university. B 1. Simplify: 1 8 9 . 1. 31. about 60 days x 53. y ≈ 18 ⋅ 1.025 doubling time of a substance or quantity is the amount of time it takes for the initial amount of that If the average number of minutes used each month is 1,160, which company offers the better plan? S0 150 53. The standard formula for area is [latex]A=LW[/latex]; however, we will solve the problem using the perimeter formula. Translate into algebra… x 3. Answer Key Chapter 6 - College Algebra | OpenStax From the graph of the function f (x) x−3 33. The half-life of a 6.5 Logarithmic Properties Let us use the car rental example above. 32 About 1.548 billion people; by the year 2031, India’s population will exceed China’s by about 0.001 b. 1. 29. 4 The domain is (−∞, 0) , the range is (−∞, ∞) , and the vertical asymptote is x also be written as the logarithm, logb x,and the value of x is the result of raising b to the power of y. 27. x = ln (3) $13,693 larger than forest A and will remain that way as long as the population growth models hold. Yes. Courses" rewritten as a single logarithm with the same base. x 11. log (7x−1) 12 4 (x+5) ; 4 ) this answer could also be written log ( (2x+3) 4 3 x (x+5) The only difference is that "College Physics" omits the "Test Prep 5 = log (11) Once this is determined, select the appropriate 9/27/2020 Answer Key Chapter 6 - College Algebra | OpenStax Try It 6.1 Exponential Functions 1. g(x) = 0.875 andj(x) = linear function. plug it into your calculator. The y-intercept on the graph of a logistic equation corresponds to the initial population for the population log(38)+5 log(3) ≈ 2.078 4 log(3) 73. x ≈ 2.2401 Specifically, it is the exponent to which a base b is raised to produce a given n ) n − 1 Then for some number −nx . The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Her speed was 5 mi/h slower on Sunday than on Saturday. A = 125e(−0.3567t) ; A ≈ 43 mg Answer: 19 2. 14/38 9/27/2020 Answer Key Chapter 6 - College Algebra | OpenStax 43. 11 b ) log (11) Final answers are Substitute the perimeter value and the expression for length into the formula. 21. f (x) = ( 1 5 5 log (11) By the quotient rule:log6 5 = a−b Domain: (−∞, ) ; Range: (−∞, ∞) = a(e) r 8. 3 e x − 2; the domain is (−∞, ∞) ; the range is (−∞, −2) ; the horizontal asymptote is y = −2. 6.4 Graphs of Logarithmic Functions 6.7 Exponential and Logarithmic Models 8 ) by reducing the fraction to lowest terms. The formula for the volume of a box is given as [latex]V=LWH[/latex], the product of length, width, and height. The pH increases by about 0.301. Regression analysis is the process of finding an equation that best fits a given set of data points. 9. x 11 2 is equivalent to 5 47. log (x) = 1.5; x ≈ 31.623 As these amounts do not change, we call them fixed costs. C 6.8 Section Exercises Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. b. Sometimes the difference between a right and wrong answer is how you 23. 45. 11. exponential; f (x) = 1.2 The length is [latex]6[/latex] in. 47. x ≈ −2.953 3. 37. f (x) = 4−x 4 e 2 3 ≈ 2.5 19/38 9/27/2020 57. x Answer Key Chapter 6 - College Algebra | OpenStax = e+10 3 15. 65. 4 ≈ 3.2 59. 57. f (x) = 1.034341(1.281204) ; g(x) = 4.035510 ; the regression curves are symmetrical about 2. 1. A table, such as the one below, is often helpful for keeping track of information in these types of problems. 2. Let b and n be positive integers greater than 1. The length is [latex]3[/latex] ft. greater than the width. 7143 grow indefinitely since resources such as food, water, and space are limited, so a logistic model best 3 18/38 9/27/2020 Answer Key Chapter 6 - College Algebra | OpenStax 47. x = 0 49. x = 51. x = 9 53. x = 55. x = −5 3 For example, the mass of Saturn is 95 11. 10. x = 1 b 23. logy 39 ( 100 ) = x 25. ln(h) = k 27. x −3 29. x 5. An order of magnitude is the nearest power of ten by which a quantity exponentially grows. The logistic regression model that fits these data is y = 25.65665979 We have two unknown quantities, length and width. 17. k 17 = log ( 6. log (123) ≈ 2.0899 = e 10. x = 0. quantity to decay. The graphs of f (x) = log We know that the length is 6 in. As both equations equal the same distance, we set them equal to each other and solve for r. We have solved for the rate of speed to work, 40 mph. 35. r ≈ −0.0667, So the hourly decay rate is about 6.67% (y) log(x) − 2 log(y) 6.3 Section Exercises 5.5556 4. MMS magnitude: 5.82 We usually interpret the word is as an equal sign. 3. Is it possible for Josh to end √ 25. log7 (15) = 27. log11 ln(15) 17. logc (k) = d n logb (x). 5. f (x) = 2(1.5) function f (x) = log(x). 37. y = 5.063 + 1.934log(x) 3 log(y) ) gives x = e2 , which is a real number. 17. 8. about $3,644,675.88 Be sure the solution can be explained in words including the units of measure. below. 11. 29. P (0) = 22 ; 175 more than the width. 17. ln(0.5) years ≈ 226, 572, 993 years. 5 )−3 8 25. no solution Consider a car rental agency that charges $0.10/mi plus a daily fee of $50. The perimeter of a rectangular outdoor patio is [latex]54[/latex] ft.