Exponential and Logarithmic Functions Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 12
Section 2.4 :: Exponential Functions 2 / 12
Minority Population According to the U.S. Census Bureau, the United States is becoming more diverse. Based on U.S. Census population projections for 2000 to 2050, the projected Hispanic population (in millions) can be modeled by the exponential function h ( t ) = 37 . 79 (1 . 021) t , where t = 0 corresponds to 2000 and 0 ≤ t ≤ 50. Source: U.S. Census Bureau Find the projected Hispanic population for 2005. Compare this to the actual value of 42.69 million. 3 / 12
Minority Population The U.S. Asian population is also growing exponentially, and the projected Asian population (in millions) can be modeled by the exponential function a ( t ) = 11 . 14 (1 . 023) t , where t = 0 corresponds to 2000 and 0 ≤ t ≤ 50. Find the projected Asian population for 2005, and compare this to the actual value of 12.69 million. 4 / 12
Minority Population Determine the expected annual percentage increase for Hispanics and for Asians. Which minority population, Hispanic or Asian, is growing at a faster rate? 5 / 12
Section 2.5 :: Logarithmic Functions 6 / 12
Interest Mary Klingman invests $ 15,000 in an account paying 7% per year compounded annually. (a) How many years are required for the compound amount to at least double? (Note that interest is only paid at the end of each year.) (b) In how many years will the amount at least triple? (c) Check your answer to part a using either the rule of 70 or the rule of 72, whichever applies. 7 / 12
Intensity of Sound A story on the National Public Radio program All Things Considered, discussed a proposal to lower the noise limit in Austin, Texas, from 85 decibels to 75 decibels. A manager for a restaurant was quoted as saying, “If you cut from 85 to 75, . . . you’re basically cutting the sound down in half.” Is this correct? If not, to what fraction of its original level is the sound being cut? Source: National Public Radio. 8 / 12
Present Value A couple wants to have $ 40,000 in 5 years for a down payment on a new house. (a) How much should they deposit today, at 6.4% compounded quarterly, to have the required amount in 5 years? (b) How much interest will be earned? (c) If they can deposit only $ 20,000 now, how much more will they need to complete the $ 40,000 after 5 years? (d) Suppose they can deposit $ 20,000 now in an account that pays interest continuously. What interest rate would they need to accumulate the entire $ 40,000 in 5 years? 9 / 12
Growth of Bacteria The growth of bacteria in food products makes it necessary to time-date some products (such as milk) so that they will be sold and consumed before the bacteria count is too high. Suppose for a certain product that the number of bacteria present is given by f ( t ) = 500 e 0 . 1 t , under certain storage conditions, where t is time in days after packing of the product and the value of f ( t ) is in millions. If the product cannot be safely eaten after the bacteria count reaches 3000 million, how long with this take? 10 / 12
Growth of Bacteria The growth of bacteria in food products makes it necessary to time-date some products (such as milk) so that they will be sold and consumed before the bacteria count is too high. Suppose for a certain product that the number of bacteria present is given by f ( t ) = 500 e 0 . 1 t , under certain storage conditions, where t is time in days after packing of the product and the value of f ( t ) is in millions. If t = 0 corresponds to January 1, what date should be placed on the product? 11 / 12
Carbon Dating A sample from a refuse deposit near the Strait of Magellan had 60% of the carbon-14 found in a contemporary living sample. How old was the sample? 12 / 12
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