The body mass index (BMI) can be expressed with the equation:
\[ \text{BMI} = k \frac{W}{H^2} \]
where:
- \( W \) is the weight of the individual,
- \( H \) is the height of the individual,
- \( k \) is the constant of proportionality.
First, we will determine the constant \( k \) using the data provided for the individual who is 177 lbs and 77 in tall:
Given:
- \( W = 177 \) lbs
- \( H = 77 \) in
- \( \text{BMI} = 20.99 \)
Substituting these values into the equation:
\[ 20.99 = k \frac{177}{77^2} \]
Calculating \( 77^2 \):
\[ 77^2 = 5929 \]
Substituting this back into the equation gives:
\[ 20.99 = k \frac{177}{5929} \]
To find \( k \), we rearrange the equation:
\[ k = 20.99 \cdot \frac{5929}{177} \]
Calculating \( \frac{5929}{177} \):
\[ \frac{5929}{177} \approx 33.487 \]
Now, multiplying by 20.99:
\[ k = 20.99 \cdot 33.487 \approx 702.85653 \]
Now that we have \( k \), we can calculate the BMI for the individual who is 71 in tall and weighs 152 lb.
Given:
- \( W = 152 \) lbs
- \( H = 71 \) in
Plugging these values back into the BMI formula:
\[ \text{BMI} = k \frac{W}{H^2} \]
Calculating \( 71^2 \):
\[ 71^2 = 5041 \]
Now substituting the values into the equation:
\[ \text{BMI} = 702.85653 \cdot \frac{152}{5041} \]
Calculating \( \frac{152}{5041} \):
\[ \frac{152}{5041} \approx 0.03015 \]
Now multiply by \( k \):
\[ \text{BMI} \approx 702.85653 \cdot 0.03015 \approx 21.15 \]
Rounding to two decimal places, we find:
\[ \text{BMI} \approx 21.15 \]
So, the BMI for an individual who is 71 inches tall and weighs 152 lbs is approximately 21.15.