320 CHAPTER 5 Exponential and Logarithmic Functions NOTE A BAC of 0.30% results in a loss of consciousness in most people. j NOTE In most states, the blood alcohol content at which a DUI citation is given is 0.08%. j Alcohol and Driving Blood alcohol concentration (BAC) is a measure of the amount of alcohol in a person’s bloodstream. A BAC of 0.04% means that a person has 4 parts alcohol per 10,000 parts blood in the body. Relative risk is defined as the likelihood of one event occurring divided by the likelihood of a second event occurring. For example, if an individual with a BAC of 0.02% is 1.4 times as likely to have a car accident as an individual who has not been drinking, the relative risk of an accident with a BAC of 0.02% is 1.4. Recent medical research suggests that the relative risk R of having an accident while driving a car can be modeled by an equation of the form = R ekx where x is the percent concentration of alcohol in the bloodstream and k is a constant. (a) Research indicates that the relative risk of a person having an accident with a BAC of 0.02% is 1.4. Find the constant k in the equation. (b) Using this value of k, what is the relative risk if the BAC is 0.17%? (c) Using this same value of k, what BAC corresponds to a relative risk of 100? (d) If the law asserts that anyone with a relative risk of 4 or more should not have driving privileges, at what concentration of alcohol in the bloodstream should a driver be arrested and charged with DUI (driving under the influence)? Solution EXAMPLE 10 (a) For a blood alcohol concentration of 0.02% and a relative risk R of 1.4, let = x 0.02 and solve for k. = = = = ≈ ⋅ R e e k k 1.4 0.02 ln1.4 ln1.4 0.02 16.82 kx k 0.02 = = R x 1.4; 0.02 Change to logarithmic form. Solve for k. (b) A BAC of 0.17% means = x 0.17. Use = k 16.82 in the equation to find the relative risk R: = = ≈ ⋅ R e e 17.5 kx 16.82 0.17 For a blood alcohol concentration of 0.17%, the relative risk R of an accident is about 17.5. That is, a person with a BAC of 0.17% is 17.5 times as likely to have a car accident as a person with no alcohol in the bloodstream. (c) A relative risk of 100 means = R 100. Use = k 16.82 in the equation = R e . kx The blood alcohol concentration x obeys = = = ≈ e x x 100 16.82 ln100 ln100 16.82 0.27 x 16.82 R e R k , 100, 16.82 kx = = = Change to logarithmic form. Solve for x. For a blood alcohol concentration of 0.27%, the relative risk R of an accident is 100. (d) A relative risk of 4 means = R 4. Use = k 16.82 in the equation = R e . kx The blood alcohol concentration x obeys = = = ≈ e x x 4 16.82 ln4 ln4 16.82 0.082 x 16.82 A driver with a BAC of 0.082% or more should be arrested and charged with DUI.
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