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NSC - Grade 12

A runner became ill during the race and was taken to the doctor. The doctor recommended that he take [tex]11.5 \, \text{mg/kg}[/tex] of an antibiotic per day. The antibiotic is available as a 150 mg capsule.

[Adapted from mwy medicalnewstoday.com]

5.2.1 If the runner has a BMI of [tex]28.6 \, \text{kg/m}^2[/tex] and is 1.65 m tall, show using calculations that 6 capsules will be the correct dosage for the runner per day.

You may use the formula:

[tex]\[ \text{BMI} = \frac{\text{Mass in kg}}{(\text{height in m})^2} \][/tex]

Sagot :

GinnyAnsweridn
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.
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