To determine the correct dosage for the runner, we will follow a step-by-step approach based on the information provided:
1. Calculate the Mass of the Runner:
We know the formula for Body Mass Index (BMI):
[tex]\[
\text{BMI} = \frac{\text{Mass in kg}}{(\text{Height in m})^2}
\][/tex]
Rearrange the formula to solve for the Mass:
[tex]\[
\text{Mass in kg} = \text{BMI} \times (\text{Height in m})^2
\][/tex]
Given:
[tex]\[
\text{BMI} = 28.6 \, \text{kg/m}^2
\][/tex]
[tex]\[
\text{Height} = 1.65 \, \text{m}
\][/tex]
Plug in the values:
[tex]\[
\text{Mass in kg} = 28.6 \times (1.65)^2
\][/tex]
[tex]\[
\text{Mass in kg} = 28.6 \times 2.7225
\][/tex]
[tex]\[
\text{Mass in kg} = 77.8635 \, \text{kg}
\][/tex]
2. Calculate the Total Daily Dosage:
The recommended antibiotic dosage is [tex]\(11.5 \, \text{mg per kg}\)[/tex].
[tex]\[
\text{Total daily dosage} = \text{Mass in kg} \times \text{Dosage per kg}
\][/tex]
[tex]\[
\text{Total daily dosage} = 77.8635 \, \text{kg} \times 11.5 \, \text{mg/kg}
\][/tex]
[tex]\[
\text{Total daily dosage} = 895.43025 \, \text{mg}
\][/tex]
3. Determine the Number of Capsules Needed:
Each capsule contains 150 mg of the antibiotic.
[tex]\[
\text{Number of capsules} = \frac{\text{Total daily dosage}}{\text{Dosage per capsule}}
\][/tex]
[tex]\[
\text{Number of capsules} = \frac{895.43025}{150}
\][/tex]
[tex]\[
\text{Number of capsules} \approx 5.9695
\][/tex]
Since the runner cannot take a fraction of a capsule, we round up to the nearest whole number:
[tex]\[
\text{Number of capsules} = 6
\][/tex]
Therefore, the correct dosage for the runner is 6 capsules per day.
