To solve this problem involving variations, we start with the formula for Body Mass Index (BMI), which varies directly as weight and inversely as the square of height. The relationship can be expressed as:
\[ \text{BMI} = k \cdot \frac{W}{H^2} \]
where \( W \) is the weight in pounds, \( H \) is the height in inches, and \( k \) is a constant.
Step 1: Find the constant \( k \)
We know:
- For a weight \( W = 144 \) lb
- For a height \( H = 60 \) in
- The BMI = 28.12
We can substitute these values into the equation to find \( k \):
\[ 28.12 = k \cdot \frac{144}{60^2} \]
Calculating \( 60^2 \):
\[ 60^2 = 3600 \]
Now substituting this into the equation gives:
\[ 28.12 = k \cdot \frac{144}{3600} \]
Now simplify \( \frac{144}{3600} \):
\[ \frac{144}{3600} = \frac{1}{25} \]
Thus, we rewrite the equation:
\[ 28.12 = k \cdot \frac{1}{25} \]
To solve for \( k \):
\[ k = 28.12 \cdot 25 = 703 \]
Step 2: Determine the BMI for the new individual
Now we need to find the BMI for another individual with:
- Weight \( W = 165 \) lb
- Height \( H = 74 \) in
Using the same formula for BMI:
\[ \text{BMI} = k \cdot \frac{W}{H^2} \]
Substituting \( k = 703 \), \( W = 165 \), and \( H = 74 \):
\[ \text{BMI} = 703 \cdot \frac{165}{74^2} \]
Calculating \( 74^2 \):
\[ 74^2 = 5476 \]
Now substitute this back into the equation:
\[ \text{BMI} = 703 \cdot \frac{165}{5476} \]
Calculating \( \frac{165}{5476} \):
\[ \frac{165}{5476} \approx 0.0301 \]
Now multiply this by \( k \):
\[ \text{BMI} = 703 \cdot 0.0301 \approx 21.13 \]
Conclusion
Thus, the BMI for the individual who is 74 inches tall and weighs 165 pounds is approximately:
\[ \text{BMI} \approx 21.13 \]
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