6.10 Functions and Their Graphs 403 81. Thorium Dating A chemical used by scientists to determine the age of rocks (see Example 9) is thorium-229 (Th-229). The exponential function used is P t P ( ) 2 , t 0 /7340 = − where P 0 represents the original amount of Th-229 present in a rock and P t( ) represents the amount of Th-229 present after t years. An archeologist finds a rock that is estimated to be about 36,700 years old. If the rock originally had 800 milligrams of Th-229 present, how many milligrams of Th-229 are present now? 25 milligrams 82. Cesium A chemical byproduct of energy production is cesium-137 (Cs-137). Like many chemicals, Cs-137 breaks down over time into other elements. The exponential function used to estimate the amount of Cs-137 remaining in a sample is P t P ( ) 2 , t 0 /30 = − where P 0 represents the original amount of Cs-137 and P t( ) represents the amount of Cs-137 present after t years. If there are 4 grams of Cs-137 present in a sample today, how many will be present in 75 years? 0.71 grams 83. Population of China In 2021, the population of China was about 1.41 billion people, and the population growth rate was about 0.1% per year. Assuming the population of China continues to grow exponentially at this rate, the expected population of China, P t( ), in billions of people, t years after 2021 is given by the exponential function P t e ( ) 1.41 .t 0.001 = Source: Census.gov a) Use this function to predict the population of China in the year 2025. 1.42 billion people b) Use this function to predict the population of China in the year 2035. 1.43 billion people 84. Population of India In 2021, the population of India was about 1.41 billion people, and the population growth rate was about 0.8% per year. Assuming the population of India continues to grow exponentially at this rate, the expected population of India, P t( ), in billions of people, t years after 2021 is given by the exponential function P t e ( ) 1.41 .t 0.008 = Source: Census.gov a) Use this function to predict the population of India in the year 2025. 1.46 billion people b) Use this function to predict the population of India in the year 2035. 1.58 billion people 85. Radioactive Decay Radium-226 is a radioactive isotope that decays exponentially at a rate of 0.0428% per year. The amount of radium-226,R, remaining after t years can be determined by the exponential function R t R e ( ) ,t 0 0.000428 = − where R0 is the original amount present. If there are originally 10 grams of radium-226, determine the amount of radium-226 remaining after 1000 years. 6.52 grams 86. Radioactive Decay Strontium-90 is a radioactive isotope that decays exponentially at a rate of 2.8% per year. The amount of strontium-90, S, remaining after t years can be determined by the exponential function S t S e ( ) ,t 0 0.028 = − where S0 is the original amount present. If there are originally 1000 grams of strontium-90, determine the amount of strontium-90 remaining after 40 years. Round your answer to the nearest gram. 326 grams Challenge Problems/Group Activities 87. The spacing of the frets on the neck of a classical guitar is determined from the equation = − d (21.9)(2) , x (20 )/12 where = x the fret number and = d the distance in centimeters of the xth fret from the bridge. d Bridge 19th fret .......... xth fret...3rd fret 2nd fret 1st fret Nut a) Determine how far the 19th fret should be from the bridge (rounded to one decimal place). 23.2 cm b) Determine how far the 4th fret should be from the bridge (rounded to one decimal place). 55.2 cm c) The distance of the nut from the bridge can be determined by letting = x 0 in the given exponential equation. Determine the distance from the nut to the bridge (rounded to one decimal place). 69.5 cm 88. Appreciation of a House A house initially cost $150,000. The value, V, of the house, in dollars, after n years if it appreciates at a constant rate of 4% per year can be determined by the function V f n( ) 150,000(1.04) .n = = a) Determine f (8) and explain its meaning. ≈ $205,285; the value of the house after 8 years is about $205,285. b) After how many years is the value of the house greater than $250,000? (Determine by trial and error.) About 14 years 89. Speed of Light Light travels at about 186,000 miles per second through space. The distance, d, in miles that light travels in t seconds can be determined by the function = d t t ( ) 186,000 . a) Light reaches the moon from Earth in about 1.3 sec. Determine the approximate distance from Earth to the moon. 241,800 mi b) Express the distance in miles, d, traveled by light in t minutes as a function of time, t. = d t t ( ) 11,160,000 c) Light travels from the sun to Earth in about 8.3 min. Determine the approximate distance from the sun to Earth. 92,628,000 mi Research Activity 90. François Viète The idea of using variables in algebraic equations was introduced by the French mathematician François Viète (1540–1603). Write a paper about his life and his contributions to mathematics. In particular, discuss his work with algebraic equations. $
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