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If 800 milligrams of medication enters a patient’s bloodstream at noon and decays exponentially at a rate of 12% per hour, when will only 8% of the original amount remain in the patient’s bloodstream? Round your answer to the nearest hundredth, if necessary.

Sagot :

Using an exponential function, it is found that it will take 19.76 hours for there to be only 8% of the original amount in the patient's bloodstream.

What is an exponential function?

A decaying exponential function is modeled by:

[tex]A(t) = A(0)(1 - r)^t[/tex]

In which:

  • A(0) is the initial value.
  • r is the decay rate, as a decimal.

For this problem, the decay rate is of 12%, hence:

r = 0.12.

And the equation is:

[tex]A(t) = A(0)(0.88)^t[/tex]

8% of the original amount will be in the patient's bloodstream after t hours, for which A(t) = 0.08A(0), hence:

[tex]A(t) = A(0)(0.88)^t[/tex]

[tex]0.08A(0) = A(0)(0.88)^t[/tex]

[tex](0.88)^t = 0.08[/tex]

[tex]\log{(0.88)^t} = \log{0.08}[/tex]

[tex]t\log{0.88} = \log{0.08}[/tex]

[tex]t = \frac{\log{0.08}}{\log{0.88}}[/tex]

t = 19.76 hours.

It will take 19.76 hours for there to be only 8% of the original amount in the patient's bloodstream.

More can be learned about exponential functions at https://brainly.com/question/25537936

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