TER 4 Inverse, Expor 78. (Modeling) Planets' Distances from the Sun and Periods of Revolution The table contains the planets' average distances D from the sun and their periods P of revolution around the sun in years. The distances have been normalized so that Earth is one unit away from the sun. For example, since Jupiter's distance is 5.2, its distance from the sun is 5.2 times farther than Earth's (a) Using a graphing calculator, make a scatter diagram by plotting the point (In D, In P) for each planet on the xy-coordinate axes. Do the data points appear to be linear? Planet O. 0.39 Mercury 0.62 O.72 Venus 1 1 Earth 1.89 1.52 Mars 11.9 5.2 Jupiter 29.5 9.54 Saturn 84.0 19.2 Uranus 164.8 30.1 Neptune ba Source: Ronan, C., The Natural History of the Universe, MacMillan Publishing Co., New York. (b) Determine a linear equation that models the data points. Graph the line and the data on the same coordinate axes. (c) Use this linear model to predict the period of Pluto if its distance is 39.5. Compare the answer to the actual value of 248.5 yr. Use the change -of-base theo rem to find an approximation to four decimal places for each logarithm. See Example 8. 79. log2 5 80. log2 9 82. logg 0.71 81. logs 0.59 83. log1/2 3 slq 84. log1/3 2 86. logV2 85. logT e 87. logV13 12 88. logV19 90. logo.91 8 89. logo.32 5 5 Let u In a and v ln b. Write each expression in terms of u and v without using the 1 In function. a3 91. In (bVa) a3 а 93. In b5 92. ln 94. In(Va b) b2 tdoo Concept Check Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)-(c). 95. Given g(x) = e*, find (a) g(In 4) (b) g(In 52) (e) 8(In) 96. Given f(x) 3, find (a) f(logs 2) (b) f(log3 (In 3)) (c) f(log3 (2 In 3)). 97. Given f(x) = In x, find (a) f(e) 98. Given f(x) = log2 x, find (a) f(2) (b) f(eln 3) (c) f(e2 In 3). (b) f(2 os 2) (c) f(22log: 2). Work each problem. 99. Concept Check Which of the following is equivaient to 2 In (3x) for x> 0? A. In 9+ In x В. In 6x C. In 6+ In x 100. Concept Check Which of the following is equivalent to In (4x)- In (2x) for x> 0 D. In 9x2 A. 2 In x In 4x С. In 2x B. In 2x D. In 2

TER 4 Inverse, Expor 78. (Modeling) Planets' Distances from the Sun and Periods of Revolution The table contains the planets' average distances D from the sun and their periods P of revolution around the sun in years. The distances have been normalized so that Earth is one unit away from the sun. For example, since Jupiter's distance is 5.2, its distance from the sun is 5.2 times farther than Earth's (a) Using a graphing calculator, make a scatter diagram by plotting the point (In D, In P) for each planet on the xy-coordinate axes. Do the data points appear to be linear? Planet O. 0.39 Mercury 0.62 O.72 Venus 1 1 Earth 1.89 1.52 Mars 11.9 5.2 Jupiter 29.5 9.54 Saturn 84.0 19.2 Uranus 164.8 30.1 Neptune ba Source: Ronan, C., The Natural History of the Universe, MacMillan Publishing Co., New York. (b) Determine a linear equation that models the data points. Graph the line and the data on the same coordinate axes. (c) Use this linear model to predict the period of Pluto if its distance is 39.5. Compare the answer to the actual value of 248.5 yr. Use the change -of-base theo rem to find an approximation to four decimal places for each logarithm. See Example 8. 79. log2 5 80. log2 9 82. logg 0.71 81. logs 0.59 83. log1/2 3 slq 84. log1/3 2 86. logV2 85. logT e 87. logV13 12 88. logV19 90. logo.91 8 89. logo.32 5 5 Let u In a and v ln b. Write each expression in terms of u and v without using the 1 In function. a3 91. In (bVa) a3 а 93. In b5 92. ln 94. In(Va b) b2 tdoo Concept Check Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)-(c). 95. Given g(x) = e*, find (a) g(In 4) (b) g(In 52) (e) 8(In) 96. Given f(x) 3, find (a) f(logs 2) (b) f(log3 (In 3)) (c) f(log3 (2 In 3)). 97. Given f(x) = In x, find (a) f(e) 98. Given f(x) = log2 x, find (a) f(2) (b) f(eln 3) (c) f(e2 In 3). (b) f(2 os 2) (c) f(22log: 2). Work each problem. 99. Concept Check Which of the following is equivaient to 2 In (3x) for x> 0? A. In 9+ In x В. In 6x C. In 6+ In x 100. Concept Check Which of the following is equivalent to In (4x)- In (2x) for x> 0 D. In 9x2 A. 2 In x In 4x С. In 2x B. In 2x D. In 2

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter3: Radian Measure

Section3.5: Velocities

Problem 68PS

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How do you implement the change-of-base theorem? Can you please provide your example answering question 79?

Transcribed Image Text:TER 4 Inverse, Expor 78. (Modeling) Planets' Distances from the Sun and Periods of Revolution The table contains the planets' average distances D from the sun and their periods P of revolution around the sun in years. The distances have been normalized so that Earth is one unit away from the sun. For example, since Jupiter's distance is 5.2, its distance from the sun is 5.2 times farther than Earth's (a) Using a graphing calculator, make a scatter diagram by plotting the point (In D, In P) for each planet on the xy-coordinate axes. Do the data points appear to be linear? Planet O. 0.39 Mercury 0.62 O.72 Venus 1 1 Earth 1.89 1.52 Mars 11.9 5.2 Jupiter 29.5 9.54 Saturn 84.0 19.2 Uranus 164.8 30.1 Neptune ba Source: Ronan, C., The Natural History of the Universe, MacMillan Publishing Co., New York. (b) Determine a linear equation that models the data points. Graph the line and the data on the same coordinate axes. (c) Use this linear model to predict the period of Pluto if its distance is 39.5. Compare the answer to the actual value of 248.5 yr. Use the change -of-base theo rem to find an approximation to four decimal places for each logarithm. See Example 8. 79. log2 5 80. log2 9 82. logg 0.71 81. logs 0.59 83. log1/2 3 slq 84. log1/3 2 86. logV2 85. logT e 87. logV13 12 88. logV19 90. logo.91 8 89. logo.32 5 5 Let u In a and v ln b. Write each expression in terms of u and v without using the 1 In function. a3 91. In (bVa) a3 а 93. In b5 92. ln 94. In(Va b) b2 tdoo Concept Check Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)-(c). 95. Given g(x) = e*, find (a) g(In 4) (b) g(In 52) (e) 8(In) 96. Given f(x) 3, find (a) f(logs 2) (b) f(log3 (In 3)) (c) f(log3 (2 In 3)). 97. Given f(x) = In x, find (a) f(e) 98. Given f(x) = log2 x, find (a) f(2) (b) f(eln 3) (c) f(e2 In 3). (b) f(2 os 2) (c) f(22log: 2). Work each problem. 99. Concept Check Which of the following is equivaient to 2 In (3x) for x> 0? A. In 9+ In x В. In 6x C. In 6+ In x 100. Concept Check Which of the following is equivalent to In (4x)- In (2x) for x> 0 D. In 9x2 A. 2 In x In 4x С. In 2x B. In 2x D. In 2

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