Questions: Body mass index (BMI) is a measurement that helps determine obesity, with a BMI of 30 or greater indicating that the person is obese. BMI is directly proportional to the weight of a person of a given height. a. If the BMI of a person who is 1.7 meters tall is 20 when the person weighs 55 kilograms, what is the constant of variation? b. If a person of this height has a BMI of 32, what do they weigh? a. The constant of variation is (Type an integer or a simplified fraction)

Solution Steps

To solve the given problem, we need to use the formula for BMI and the concept of direct proportionality. The BMI formula is:

\[ \text{BMI} = \frac{\text{weight (kg)}}{\text{height (m)}^2} \]

a. To find the constant of variation (k), we use the given BMI and weight for the person of height 1.7 meters. b. To find the weight of a person with a BMI of 32 and height 1.7 meters, we use the constant of variation found in part (a).

1. Use the given BMI and weight to find the constant of variation (k).
2. Use the constant of variation to find the weight for the given BMI.

Given:

• Height \( h = 1.7 \) meters
• Weight \( w = 55 \) kilograms
• BMI \( \text{BMI} = 20 \)

The formula for BMI is: \[ \text{BMI} = \frac{w}{h^2} \]

To find the constant of variation \( k \): \[ k = \frac{\text{BMI}}{\frac{w}{h^2}} \]

Substituting the given values: \[ k = \frac{20}{\frac{55}{1.7^2}} \] \[ k \approx 1.0509 \]

Given:

• New BMI \( \text{BMI}_{\text{new}} = 32 \)
• Height \( h = 1.7 \) meters

Using the constant of variation \( k \) found in Step 1, we can find the new weight \( w_{\text{new}} \): \[ w_{\text{new}} = \text{BMI}_{\text{new}} \times h^2 \]

Substituting the given values: \[ w_{\text{new}} = 32 \times 1.7^2 \] \[ w_{\text{new}} \approx 92.48 \text{ kg} \]

Final Answer