To determine the correct dosage of the antibiotic in terms of the number of capsules the runner should take per day, we need to follow a systematic approach based on the provided formula and given values.
### Step 1: Calculate the Mass of the Runner
Given:
- Body Mass Index (BMI) = [tex]\( 28.6 \, \text{kg} / \text{m}^2 \)[/tex]
- Height ([tex]\( h \)[/tex]) = 1.65 m
The formula for BMI is:
[tex]\[ \text{BMI} = \frac{\text{Mass (in kg)}}{(\text{Height (in m)})^2} \][/tex]
We can rearrange this formula to solve for the mass:
[tex]\[ \text{Mass (in kg)} = \text{BMI} \times (\text{Height (in m)})^2 \][/tex]
Substituting the given 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]
So, the mass of the runner is approximately [tex]\( 77.8635 \, \text{kg} \)[/tex].
### Step 2: Calculate the Total Required Dosage per Day
Given:
- Dosage required per kg = [tex]\( 11.5 \, \text{mg} / \text{kg} \)[/tex]
We multiply this dosage by the runner's mass to find the total daily dosage:
[tex]\[ \text{Total daily dosage (in mg)} = \text{Mass (in kg)} \times \text{Dosage per kg} \][/tex]
[tex]\[ \text{Total daily dosage (in mg)} = 77.8635 \, \text{kg} \times 11.5 \, \text{mg} / \text{kg} \][/tex]
[tex]\[ \text{Total daily dosage (in mg)} = 895.43025 \, \text{mg} \][/tex]
So, the total required dosage of the antibiotic for the runner per day is approximately [tex]\( 895.43025 \, \text{mg} \)[/tex].
### Step 3: Calculate the Number of Capsules Needed
Given:
- Each capsule contains [tex]\( 150 \, \text{mg} \)[/tex] of the antibiotic.
To find the number of capsules the runner needs to take:
[tex]\[ \text{Number of capsules} = \frac{\text{Total daily dosage (in mg)}}{\text{Dosage per capsule (in mg)}} \][/tex]
[tex]\[ \text{Number of capsules} = \frac{895.43025 \, \text{mg}}{150 \, \text{mg}} \][/tex]
[tex]\[ \text{Number of capsules} \approx 5.97 \][/tex]
Since the runner cannot take a fraction of a capsule, we round up to the next whole number:
[tex]\[ \text{Number of capsules} = 6 \][/tex]
Therefore, the runner needs to take 6 capsules of the antibiotic per day to meet the required dosage.
