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 179-lb person who is 68 in, tall is 27.21. Determine the BMI for an individual who is 60 in, tall and 128 lb.

Answer 1

To solve for the unknown value using a variation model, we start with the formula for Body Mass Index (BMI):

$$\text{BMI}=k\cdot \frac{W}{H^2}$$

where $W$ is the weight in pounds, $H$ is the height in inches, and $k$ is a constant of proportionality.

Step 1: Find the constant $k$

We know the BMI for a 179-lb person who is 68 in tall is 27.21. Plugging these values into the equation:

$$27.21=k\cdot \frac{179}{{68}^2}$$

Calculating ${68}^2$:

$${68}^2=4624$$

Now substitute this into the equation:

$$27.21=k\cdot \frac{179}{4624}$$

To isolate $k$, we rearrange the equation:

$$k=27.21\cdot \frac{4624}{179}$$

Calculating $\frac{4624}{179}$:

$$\frac{4624}{179}\approx 25.8$$

Now, we multiply:

$$k\approx 27.21\cdot 25.8\approx 702.798$$

Step 2: Use $k$ to find the BMI for the 128-lb person who is 60 in tall

Now we use the constant $k$ to find the BMI for the individual who weighs 128 lb and is 60 in tall.

$$\text{BMI}=702.798\cdot \frac{128}{{60}^2}$$

Calculating ${60}^2$:

$${60}^2=3600$$

Now substitute this back into the equation:

$$\text{BMI}=702.798\cdot \frac{128}{3600}$$

Calculating $\frac{128}{3600}$:

$$\frac{128}{3600}\approx 0.03556$$

Now, substitute this into the equation for BMI:

$$\text{BMI}\approx 702.798\cdot 0.03556\approx 24.96$$

Conclusion

The body mass index (BMI) for an individual who is 60 inches tall and weighs 128 pounds is approximately:

$$\text{BMI}\approx 24.96$$