The Body Mass Index (BMI) can be expressed with the following formula, based on the problem statement:
\[ \text{BMI} = k \cdot \frac{w}{h^2} \]
where:
- \( w \) is the weight in pounds,
- \( h \) is the height in inches,
- \( k \) is a constant of variation.
Step 1: Determine the constant \( k \)
For the individual weighing 138 lb and 74 in tall with a BMI of 17.72, we can substitute the known values into the formula:
\[ 17.72 = k \cdot \frac{138}{74^2} \]
First, calculate \( 74^2 \):
\[ 74^2 = 5476 \]
Now substitute this into the equation:
\[ 17.72 = k \cdot \frac{138}{5476} \]
Next, calculate \( \frac{138}{5476} \):
\[ \frac{138}{5476} \approx 0.025187 \]
Now, substitute this back into the equation:
\[ 17.72 = k \cdot 0.025187 \]
To find \( k \), divide both sides by \( 0.025187 \):
\[ k = \frac{17.72}{0.025187} \approx 702.79 \]
Step 2: Use \( k \) to find the BMI for the new individual
Now we will use the constant \( k \) to find the BMI for an individual who is 123 lb and 73 in tall.
Substituting into the BMI formula gives:
\[ \text{BMI} = 702.79 \cdot \frac{123}{73^2} \]
Calculate \( 73^2 \):
\[ 73^2 = 5329 \]
Now substitute back into the equation:
\[ \text{BMI} = 702.79 \cdot \frac{123}{5329} \]
First, calculate \( \frac{123}{5329} \):
\[ \frac{123}{5329} \approx 0.023063 \]
Now plug this into the equation:
\[ \text{BMI} = 702.79 \cdot 0.023063 \approx 16.21 \]
Step 3: Conclusion
The BMI for an individual who is 73 inches tall and weighs 123 pounds is approximately:
\[ \boxed{16.21} \]