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 1.6 meters tall is 24 when the person weighs 66 ​kilograms, what is the constant of​ variation?
b. If a person of this height has a BMI of 36​, what do they​ weigh?

### Part (a): Finding the Constant of Variation

BMI is given by the formula:

\[
\text{BMI} = k \times \text{weight}
\]

where \( k \) is the constant of variation, and weight is in kilograms. Since BMI is also inversely proportional to the square of height, the formula can be expressed as:

\[
\text{BMI} = \frac{k \times \text{weight}}{\text{height}^2}
\]

Given:
- BMI = 24
- Height = 1.6 meters
- Weight = 66 kilograms

Substituting the known values into the equation:

\[
24 = \frac{k \times 66}{(1.6)^2}
\]

Now, solve for \( k \):

\[
24 = \frac{66k}{2.56}
\]

\[
66k = 24 \times 2.56
\]

\[
66k = 61.44
\]

\[
k = \frac{61.44}{66} \approx 0.9309
\]

So, the constant of variation \( k \) is approximately **0.9309**.

### Part (b): Finding the Weight

Now, use the constant of variation \( k \) to find the weight of a person with a BMI of 36.

Given:
- BMI = 36
- Height = 1.6 meters
- \( k \approx 0.9309 \)

We can use the formula:

\[
\text{BMI} = \frac{k \times \text{weight}}{\text{height}^2}
\]

Substitute the values:

\[
36 = \frac{0.9309 \times \text{weight}}{2.56}
\]

Solve for the weight:

\[
36 \times 2.56 = 0.9309 \times \text{weight}
\]

\[
92.16 = 0.9309 \times \text{weight}
\]

\[
\text{weight} = \frac{92.16}{0.9309} \approx 99 \text{ kilograms}
\]

So, the person would weigh approximately **99 kilograms**.

---

Would you like further details or have any questions?

Here are some related questions:
1. How does the BMI formula change if height is in centimeters instead of meters?
2. What would be the weight of a person with a BMI of 30 and a height of 1.7 meters?
3. How does the constant of variation change with different heights?
4. What is the interpretation of BMI values between 25 and 29.9?
5. How does body composition (muscle vs. fat) affect the accuracy of BMI as an obesity indicator?

**Tip:** Remember that BMI does not distinguish between muscle and fat, so athletes may have a high BMI but low body fat percentage.

Learn how to calculate BMI and interpret its results with examples. Solve problems related to BMI calculations, including finding the constant of variation and determining weight based on BMI values. Suitable for high school students.

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