500 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions 111. CO2 Emissions Tax One action that government could take to reduce carbon emissions into the atmosphere is to levy a tax on fossil fuel. This tax would be based on the amount of carbon dioxide emitted into the air when the fuel is burned. The cost-benefit equation ln 11 - P2 = -0.0034 - 0.0053x models the approximate relationship between a tax of x dollars per ton of carbon and the corresponding percent reduction P (in decimal form) of emissions of carbon dioxide. (Data from Nordhause, W., “To Slow or Not to Slow: The Economics of the Greenhouse Effect,” Yale University, New Haven, Connecticut.) (a) Write P as a function of x. (b) Graph P for 0 … x … 1000. Discuss the benefit of continuing to raise taxes on carbon. (c) Determine P, to the nearest tenth, when x = $60. Interpret this result. (d) What value of x will give a 50% reduction in carbon emissions? 112. Radiative Forcing Radiative forcing, R, measures the influence of carbon dioxide in altering the additional solar radiation trapped in Earth’s atmosphere. The International Panel on Climate Change (IPCC) in 1990 estimated k to be 6.3 in the radiative forcing equation R = k ln C C0 , where C0 is the preindustrial amount of carbon dioxide and C is the current level. (Data from Clime, W., The Economics of Global Warming, Institute for International Economics, Washington, D.C.) (a) Use the equation R = 6.3 ln C C0 to determine the radiative forcing R, in watts per square meter 1W/m22 to the nearest tenth, expected by the IPCC if the carbon dioxide level in the atmosphere doubles from its preindustrial level. (b) Determine the global temperature increase T, in degrees Fahrenheit to the nearest tenth, that the IPCC predicted would occur if atmospheric carbon dioxide levels were to double, given T1R2 = 1.03R. Find ƒ -11x2, and give the domain and range. 113. ƒ1x2 = ex-5 114. ƒ1x2 = ex + 10 115. ƒ1x2 = ex+1 - 4 116. ƒ1x2 = ln 1x + 22 117. ƒ1x2 = 2 ln 3x 118. ƒ1x2 = ln 1x - 12 + 6 Use a graphing calculator to solve each equation. Give irrational solutions correct to the nearest hundredth. 119. ex + ln x = 5 120. ex - ln 1x + 12 = 3 121. 2ex + 1 = 3e-x 122. ex + 6e-x = 5 123. log x = x2 - 8x + 14 124. ln x = -23 x + 3 125. Find the error in the following “proof” that 2 61. 1 9 6 1 3 True statement a1 3b 2 6 1 3 Rewrite the left side. log a 1 3b 2 6log 1 3 Take the logarithm on each side. 2 log 1 3 61 log 1 3 Property of logarithms; identity property 2 61 Divide each side by log 1 3 .
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