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?

To find out how much of the antibiotic is in Han's body right after the third dose, let's break down the calculations step by step: 1. **First Dose (Day 1)**: Han takes 150 mg at time \( t = 0 \). After 12 hours, 5% of the dose remains. - Remaining after 12 hours = \( 150 \times 0.05 = 7.5 \) mg. 2. **Second Dose (Day 1)**: Han takes another 150 mg at \( t = 12 \) hours. Now, the total amount of the antibiotic is: - Total = Previous remaining (7.5 mg) + New dose (150 mg) = \( 7.5 + 150 = 157.5 \) mg. - After another 12 hours, only 5% of the new dose remains, and we need to consider the decay of the first dose too. - Remaining first dose after the second 12 hours = \( 150 \times 0.05 = 7.5 \) mg; however, since more time has passed, that will decay: - Total after second dose (still 12 hrs later) = \( 157.5 \times 0.05 = 7.875 \) mg. 3. **Third Dose (Day 2)**: Han takes another 150 mg at \( t = 24 \) hours. Total now is: - New Total = Remaining from the first dose + Remaining from the second dose + New dose = \( 7.5 + 7.875 + 150 = 165.375 \) mg. - After 12 hours, only 5% remains of the new dose as well. - After the 12 hours (total 36 hours after beginning): \( 165.375 \times 0.05 = 8.26875 \) mg still remains at 36 hours plus the last dose added. Calculating this could be a bit long, but you can use the exponential decay formula applied to each dose and accumulate how many times the 5% remains. 4. **Final Calculation**: Right after the third dose, before decay, Han has: - \( 165.375 + 150 = 315.375 \) mg just after the third dose, considering all previous doses as decayed. After combining and reducing: - Just right after the 2nd plus before decay = \( 165 + continuing 150 = 315.375\)mg. Therefore, right after the third dose, Han has approximately **315.4 mg** of the antibiotic in his body.