131. A patient in a hospital needs to maintain a certain amount of a medication in her bloodstream to fight an infection. Suppose the initial dosage is 10 mg, and the patient is given an additional maintenance dose of 4 mg every hour. Assume that the amount of medication in the bloodstream is reduced by 25% every hour.
a. Write a function for the amount of the initial dosage that remains in the bloodstream after n hours.

where Y, is the value in milligrams of the patient's dose, b is the intercept with the Y axis and x is the number of hours that pass, and m is the slope, taking into account the above we infer the following equation

Y=mn+b

Now what we do is find every 2 points of the line using the problem data, in order to find the complete equation and the slope,

N1=0 Y1=10mg

N2=1 Y2=10mg+4mg=14mg

b is the value of Y when n = 0 therefore b = 10

Now we remember the equation of the slope

[tex]m=\frac{Y2-Y1}{N2-N1} =\frac{14-10}{1-0} =4mg/hour[/tex]

the ecuation is

Y=4n+10

Finally, the problem explains that the body loses 25% of the medication, therefore we multiply the equation by the 75% that remains in the body taking into account that the initial dose for N = 0 must be 10mg.

[tex]Y=0.75(4n+\frac{10}{0.75} )=3n+10\\\\[/tex]

final ecuation

Y=3n+10