Question

Sophia has an ear infection. The doctor prescribes a course of antibiotics. Sophia is told to take 500 mg doses of the antibiotic regularly every 12 hours for 10 days.
Sophia is curious and wants to know how much of the drug will be in her body over the course of the 10 days. She does some research online and finds out that at the end of 12 hours, about 4.5% of the drug is still in the body.

What quantity of the drug is in the body right after the first dose, the second dose, the third dose, the fourth dose?

When will the total amount of the antibiotic in Sophia’s body be the highest? What is that amount?

Answer Sophia’s original question: Describe how much of the drug will be in her body at various points over the course of the 10 days?

Answer 1

Part 1:

Let 

      Q₁ = Amount of the drug in the body after the first dose.
      Q₂ =  250 mg

As we know that after 12 hours about 4% of the drug is still present in the body.
For Q₂,
we get:
             Q₂ = 4% of Q₁ + 250
                   = (0.04 × 250) + 250
                   = 10 + 250
                   = 260 mg

Therefore, after the second dose, 260 mg of the drug is present in the body.

Now, for Q₃ :

We get;
           Q₃ = 4% of Q2 + 250
                = 0.04 × 260 + 250
                = 10.4 + 250
                = 260.4

For Q₄,
We get;
           Q₄ = 4% of Q₃ + 250 
                 = 0.04 × 260.4 + 250
                 = 10.416 + 250 
                 = 260.416

Part 2:
To find out how large that amount is, we have to find Q₄₀.
Using the similar pattern

for Q₄₀,
We get;
            Q₄₀ = 250 + 250 × (0.04)¹ + 250 × (0.04)² + 250 × (0.04)³⁹

Taking 250 as common;

            Q₄₀ = 250 (1 + 0.04 + 0.042 + ⋯ + 0.0439)
                  = 2501 − 0.04401 − 0.04
            Q₄₀ = 260.4167

Hence, The greatest amount of antibiotics in Susan’s body is 260.4167 mg.

Part 3:
From the previous 2 components of the matter, we all know that the best quantity of the antibiotic in Susan's body is regarding 260.4167 mg and it'll occur right once she has taken the last dose. However, we have a tendency to see that already once the fourth dose she had 260.416 mg of the drug in her system, that is simply insignificantly smaller. thus we will say that beginning on the second day of treatment, double every day there'll be regarding 260.416 mg of the antibiotic in her body. Over the course of the subsequent twelve hours {the quantity|the quantity|the number} of the drug can decrease to 4% of the most amount, that is 10.4166 mg. Then the cycle can repeat.