SECTION 5.3 Exponential Functions 305 Solving an Exponential Equation Solve: ( ) = ⋅ −e e e 1 x x 2 3 2 Solution EXAMPLE 8 Use the Laws of Exponents first to get a single expression with the base e on the right side. ( ) ⋅ = ⋅ = − − e e e e e 1 x x x 2 3 2 3 2 3 Then, e e x x x x x x x x 2 3 2 3 0 3 1 0 3 or 1 x x2 3 2 2 2 ( )( ) = − = − + − = + − = = − = − − The solution set is { } −3, 1 . Use property (3). Place the quadratic equation in standard form. Factor. Use the Zero-Product Property. Now Work PROBLEM 85 Exponential Probability Between 9:00 pm and 10:00 pm, cars arrive at Burger King’s drive-thru at the rate of 12 cars per hour (0.2 car per minute).The following formula from probability theory can be used to determine the probability that a car will arrive within t minutes of 9:00 pm. ( ) = − − F t e 1 t 0.2 (a) Determine the probability that a car will arrive within 5 minutes of 9 pm (that is, before 9:05 pm). (b) Determine the probability that a car will arrive within 30 minutes of 9 pm (before 9:30 pm). (c) Graph F. (d) What does F approach as t increases without bound in the positive direction? EXAMPLE 9 Figure 33 ( ) F 5 Figure 34 ( ) = − − F t e 1 t 0.2 0 1 0 30 Solution (a) The probability that a car will arrive within 5 minutes is found by evaluating ( ) F t at = t 5. ( ) = − ≈ − ⋅ F e 5 1 0.63212 0.2 5 ↑ Use a calculator. See Figure 33. There is a 63% probability that a car will arrive within 5 minutes. (b) The probability that a car will arrive within 30 minutes is found by evaluating ( ) F t at = t 30. ( ) = − ≈ − ⋅ F e 30 1 0.9975 0.2 30 ↑ Use a calculator. There is a 99.75% probability that a car will arrive within 30 minutes. (c) See Figure 34 for the graph of F using a TI-84 Plus CE. (d) As time passes, the probability that a car will arrive increases. The value that F approaches can be found by letting →∞ t . Since = −e e 1 , t t 0.2 0.2 it follows that → −e 0 t 0.2 as →∞ t . Therefore, ( ) = − → →∞ − F t e t 1 1 as . t 0.2 The algebraic analysis is supported by Figure 34. Now Work PROBLEM 117
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