Refer to the following: During the course of treatment of an...

Refer to the following: During the course of treatment of an illness, the concentration of a drug in the bloodstream in micrograms per microliter fluctuates during the dosing period of 8 hours according to the model $$ C(t)=15.4-4.7 \sin \left(\frac{\pi}{4} t+\frac{\pi}{2}\right), \quad 0 \leq t \leq 8 $$ Note: This model does not apply to the first dose of the medication. Find the concentration of the drug in the bloodstream 6 hours after taking a dose of the drug.

Refer to the following:
During the course of treatment of an illness, the concentration of a drug in the bloodstream in micrograms per microliter fluctuates during the dosing period of 8 hours according to the model
$$
C(t)=15.4-4.7 \sin \left(\frac{\pi}{4} t+\frac{\pi}{2}\right), \quad 0 \leq t \leq 8
$$
Note: This model does not apply to the first dose of the medication.
Find the concentration of the drug in the bloodstream 6 hours after taking a dose of the drug.

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Key Concepts

Sinusoidal Functions

Sinusoidal functions, including sine and cosine, are fundamental in representing periodic or oscillatory behavior. They are characterized by parameters such as amplitude, period, phase shift, and vertical shift, which together determine the shape and positioning of the function's graph. These functions are essential in modeling phenomena that exhibit regular fluctuation over time.

Function Evaluation

Function evaluation involves substituting a specific input value into a mathematical function to determine the corresponding output. This concept is crucial in many areas of mathematics and applied sciences, as it allows for the precise prediction of a system's behavior at a particular point in time.

Mathematical Modeling

Mathematical modeling is the process of representing real-world situations through mathematical constructs such as equations and functions. This approach enables complex systems, like those found in biological or pharmaceutical contexts, to be analyzed, predicted, and understood using abstract mathematical principles.

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