Han is prescribed a course of antibiotics for an infection. He is told to take a 150 mg dose of the antibiotic regularly every 12 hours for 15 days. Han is curious about the antibiotic and learns that at the end of the 12 hours, only $$ 5 \% $$ of the dose is still in his body. How much of the antibiotic is in the body right after the third dose?

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UpStudy AI · Accepted Answer

Let the amount in the body immediately after the $$ n $$th dose be $$ A_n $$. Each dose is 150 mg. Before each new dose, only $$ 5\% $$ of the previous amount remains, so

$$
A_n = 150 + 0.05 \, A_{n-1}.
$$

We start with the first dose:

1. After the first dose, 
   $$
   A_1 = 150.
   $$

2. After 12 hours, only $$ 5\% $$ of $$ A_1 $$ remains:
   $$
   0.05 \times 150 = 7.5 \text{ mg}.
   $$
   When the second dose is taken, the amount is:
   $$
   A_2 = 150 + 7.5 = 157.5.
   $$

3. After the next 12 hours, the leftover from $$ A_2 $$ is:
   $$
   0.05 \times 157.5 = 7.875 \text{ mg}.
   $$
   Adding the third dose gives:
   $$
   A_3 = 150 + 7.875 = 157.875.
   $$

Thus, immediately after the third dose, the amount of antibiotic in the body is

$$
\boxed{157.875 \text{ mg}}.
$$
After the third dose, there are 157.875 mg of antibiotic in Han's body.

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