Use a variation model to solve for the unknown value.

Question

The body mass index (BMI) of an individual varies directly as the weight of the individual and inversely as the square of the height of the individual. The body mass index for a 138-lb person who is 74 in. tall is 17.72. Determine the BMI for an individual who is 73 in. tall and 123 lb.

Answer 1

The Body Mass Index (BMI) can be expressed with the following formula, based on the problem statement:

$$\text{BMI}=k\cdot \frac{w}{h^2}$$

where:

• $w$ is the weight in pounds,
• $h$ is the height in inches,
• $k$ is a constant of variation.

Step 1: Determine the constant $k$

For the individual weighing 138 lb and 74 in tall with a BMI of 17.72, we can substitute the known values into the formula:

$$17.72=k\cdot \frac{138}{{74}^2}$$

First, calculate ${74}^2$:

$${74}^2=5476$$

Now substitute this into the equation:

$$17.72=k\cdot \frac{138}{5476}$$

Next, calculate $\frac{138}{5476}$:

$$\frac{138}{5476}\approx 0.025187$$

Now, substitute this back into the equation:

$$17.72=k\cdot 0.025187$$

To find $k$, divide both sides by $0.025187$:

$$k=\frac{17.72}{0.025187}\approx 702.79$$

Step 2: Use $k$ to find the BMI for the new individual

Now we will use the constant $k$ to find the BMI for an individual who is 123 lb and 73 in tall.

Substituting into the BMI formula gives:

$$\text{BMI}=702.79\cdot \frac{123}{{73}^2}$$

Calculate ${73}^2$:

$${73}^2=5329$$

Now substitute back into the equation:

$$\text{BMI}=702.79\cdot \frac{123}{5329}$$

First, calculate $\frac{123}{5329}$:

$$\frac{123}{5329}\approx 0.023063$$

Now plug this into the equation:

$$\text{BMI}=702.79\cdot 0.023063\approx 16.21$$

Step 3: Conclusion

The BMI for an individual who is 73 inches tall and weighs 123 pounds is approximately:

$$\boxed{{\displaystyle 16.21}}$$