356 CHAPTER 5 Exponential and Logarithmic Functions Figure 55 = Y e 100 x 1 0.045 250 0 0 20 Bacterial Growth A colony of bacteria that grows according to the law of uninhibited growth is modeled by the function ( ) = N t e 100 , t 0.045 where N is measured in grams and t is measured in days. (a) Determine the initial amount of bacteria. (b) What is the growth rate of the bacteria? (c) Graph the function using a graphing utility. (d) What is the population after 5 days? (e) How long will it take for the population to reach 140 grams? (f) What is the doubling time for the population? Solution EXAMPLE 1 (a) The initial amount of bacteria, N ,0 is obtained when = t 0, so ( ) = = = ⋅ N N e 0 100 100 grams 0 0.045 0 (b) Compare ( ) = N t e 100 t 0.045 to ( ) = N t N e . kt 0 The value of k, 0.045, indicates a growth rate of 4.5%. (c) Figure 55 shows the graph of ( ) = N t e 100 t 0.045 on a TI-84 Plus CE. (d) The population after 5 days is ( ) = ≈ ⋅ N e 5 100 125.2 grams. 0.045 5 (e) To find how long it takes for the population to reach 140 grams, solve the equation ( ) = N t 140. = = = = ≈ e e t t 100 140 1.4 0.045 ln1.4 ln1.4 0.045 7.5 days t t 0.045 0.045 Divide both sides of the equation by 100. Rewrite as a logarithm. Divide both sides of the equation by 0.045. The population reaches 140 grams in about 7.5 days. (f) The population doubles when ( ) = N t 200 grams, so the doubling time is found by solving the equation = e 200 100 t 0.045 for t. = = = = ≈ e e t t 200 100 2 ln2 0.045 ln2 0.045 15.4 days t t 0.045 0.045 Divide both sides of the equation by 100. Rewrite as a logarithm. Divide both sides of the equation by 0.045. The population doubles approximately every 15.4 days. Now Work PROBLEM 1 Bacterial Growth A colony of bacteria increases according to the law of uninhibited growth. (a) If N is the number of cells and t is the time in hours, express N as a function of t. (b) If the number of bacteria doubles in 3 hours, find the function that gives the number of cells in the culture. (c) How long will it take for the size of the colony to triple? (d) How long will it take for the population to double a second time (that is, to increase four times)? EXAMPLE 2
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