The results of an experiment show that the number of bacteria doubles each hour when food is left out at 90∘∘F. If the experiment starts with one bacterium, which of the following does NOT represent the results of the experiment? f(t)=2tf(t)=2^t

Question
Functions
asked 2021-02-19
The results of an experiment show that the number of bacteria doubles each hour when food is left out at 90∘∘F. If the experiment starts with one bacterium, which of the following does NOT represent the results of the experiment?
\(\displaystyle{f{{\left({t}\right)}}}={2}{t}{f{{\left({t}\right)}}}={2}^{{t}}\)

Answers (1)

2021-02-20
If the number of bacteria is doubled in each hour and starts with 1 bacterium, then after 1 hour there will be 1(2)=2 bacteria, after 2 hours there will be 2(2)=4 bacteria, after 3 hours there will be 2(4)=8 bacteria, etc.
f(t)=2t is a linear function with a slope of 2. Linear functions have a constant rate of change which means they increase by a constant amount, not a constant factor. Since the bacteria are doubling each hour, they are increasing by a constant factor of 2. Therefore, f(t)=2t does not represent the results of the experiment.
\(\displaystyle{f{{\left({t}\right)}}}={2}^{{t}}\) is an exponential function with a base of 2. Exponential functions have a constant growth or decay factor which means they increase or decrease by a constant factor. The growth or decay factor is the base so since f(t)=2t has a base of 2, it means the bacteria is increasing by a constant factor of 2. Since \(\displaystyle{f{{\left({0}\right)}}}={2}^{{0}}={1}\), then \(\displaystyle{f{{\left({t}\right)}}}={2}^{{t}}\) also means the experiment starts with 1 bacterium. Therefore, \(\displaystyle{f{{\left({t}\right)}}}={2}^{{t}}\) does represent the results of the experiment.
There are 60 minutes in 1 hour so if the number of bacteria is 1 after 0 hours = 0 minutes, then the graph must pass through (0,1). If the number of bacteria is 2 after 1 hour = 60 minutes, then the graph must pass through (60,2). If the number of bacteria is 4 after 2 hours = 120 minutes, then the graph must pass through (120,4). Continuing this pattern, the graph would also need to pass through the points (180,8), (240,16), and (300,32). The given graph passes through all of these points so it does represent the results of the experiment.
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