Answer:
The function is the difference between a constant and an exponential function, the exponential function approaches to 0 as t increases. So the amount of the medication is 16 minus a quantity that is ever decreasing without take negative values. Then, that difference will ever be a number minor than 16.
Step-by-step explanation:
The problem requires to write a function for the statement. The statement is about amount of medication that remains in the bloodstream each hour "t" which we going to label as "y(t)".
The statement says that there are an initial dosage of 10 mg and that the amount of medication in the bloodstream is reduced by 25% every hour. With that data we can estimate the amount of the initial dosage that remains after "t" hours. For that we need to take into account that when an original amount is reduced by a consistent rate over a period of time, as in this case, exponential decay is occurring. Exponential decay function can be written as follows:
with y= quantity that remains, A=initial quantity, b= percentage change (in decimal form), t is the variable time.
In this case A=10mg and b=25% (0.25 in decimal form) we can write
Besides, the statement says that the patient is given an additional maintenance dose of 4 mg every hour, and again that dosage also would be reduced 25% each hour. So we have that
hours amount of maintanance dosage that remains
0 0
1 4
2 4(1+0.75)
3 4(1+0.75+0.75^2)
4 4(1+0.75+0.75^2+0.75^3)
5 4(1+0.75+0.75^2+0.75^3+0.75^4)
m 4(1+0.75+0.75^2+0.75^3+0.75^4+...+0.75^(m-1))
Note that we have a sum of terms, so we can write as partial sum as follows
In series tables you can see that this partial sum is equals to a function
So the complete function for the medicament that remains in the blood is
Now we should reorganize the function y as follows