Answer:
The 5-hour growth/decay factor for the number of mg of caffeine in Danny's body is 0.13005
Step-by-step explanation:
Using an exponential function, it is found that the decay factor after 5 hours is of 0.1397.
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An exponential function for the amount of caffeine in Danny's blood after t hours is given by:
[tex]A(t) = A(0)e^{-kt}[/tex]
In which:
The 10-hour decay factor for the number of mg of caffeine in Danny's body is 0.2601, thus:
[tex]A(10) = (1 - 0.2601)A(0) = 0.7399A(0)[/tex]
We use this to find k.
[tex]A(t) = A(0)e^{-kt}[/tex]
[tex]0.7399A(0) = A(0)e^{-10k}[/tex]
[tex]e^{-10k} = 0.7399[/tex]
[tex]\ln{e^{-10k}} = \ln{0.7399}[/tex]
[tex]-10k = \ln{0.7399}[/tex]
[tex]k = -\frac{\ln{0.7399}}{10}[/tex]
[tex]k = 0.0301[/tex]
Thus:
[tex]A(t) = A(0)e^{-0.0301t}[/tex]
After 5 hours, the amount is:
[tex]A(5) = A(0)e^{-0.0301(5)} = 0.8603[/tex]
Thus, the decay factor is of:
[tex]1 - 0.8603 = 0.1397[/tex]
A similar problem is given at https://brainly.com/question/14773454
