To solve for the unknown value of the body mass index (BMI) using the given relationships, we will use the concept of variation. The formula for BMI can be represented as:
\[ \text{BMI} = k \cdot \frac{w}{h^2} \]
where:
- \( k \) is a constant,
- \( w \) is the weight of the individual in pounds,
- \( h \) is the height of the individual in inches.
Step 1: Find the constant \( k \)
From the given information for the first individual:
- Weight (\( w \)) = 163 lb
- Height (\( h \)) = 70 in
- BMI = 23.39
We can substitute these values into the formula to find \( k \):
\[ 23.39 = k \cdot \frac{163}{70^2} \]
Calculating \( 70^2 \):
\[ 70^2 = 4900 \]
Now substituting this value in:
\[ 23.39 = k \cdot \frac{163}{4900} \]
To isolate \( k \), multiply both sides by \( \frac{4900}{163} \):
\[ k = 23.39 \cdot \frac{4900}{163} \]
Calculating \( \frac{4900}{163} \):
\[ \frac{4900}{163} \approx 30.06 \]
Now calculate \( k \):
\[ k = 23.39 \cdot 30.06 \approx 703.53 \]
Step 2: Use \( k \) to find the BMI for the second individual
Now we need to find the BMI for another individual with:
- Weight (\( w \)) = 158 lb
- Height (\( h \)) = 67 in
Using the formula, we can now substitute \( k \), \( w \), and \( h \):
\[ \text{BMI} = 703.53 \cdot \frac{158}{67^2} \]
Calculating \( 67^2 \):
\[ 67^2 = 4489 \]
Substituting this value in:
\[ \text{BMI} = 703.53 \cdot \frac{158}{4489} \]
Calculating \( \frac{158}{4489} \):
\[ \frac{158}{4489} \approx 0.0352 \]
Now substituting back to get BMI:
\[ \text{BMI} = 703.53 \cdot 0.0352 \approx 24.79 \]
Final answer
The BMI for an individual who is 67 inches tall and weighs 158 pounds is approximately:
\[ \text{BMI} \approx 24.79 \]