Certainly! Let's walk through the steps to solve the inequality for a person's healthy weight range, based on their height and Body Mass Index (BMI):
1. Understanding the Inequality:
The inequality given is:
[tex]\[
18.5 \leq \frac{703 W}{74^2} \leq 24.9
\][/tex]
where:
- [tex]\( W \)[/tex] represents the weight in pounds.
- 74 inches is the person's height.
2. Isolate [tex]\( W \)[/tex]:
To find the weight range, we need to solve for [tex]\( W \)[/tex] in the inequality. We’ll start by isolating [tex]\( W \)[/tex] in the given inequality.
3. Expand and Isolate Step-by-Step:
The inequality can be split into two parts:
[tex]\[
18.5 \leq \frac{703W}{5476} \quad \text{and} \quad \frac{703W}{5476} \leq 24.9
\][/tex]
4. Calculate Height Squared:
First, find the height squared:
[tex]\[
74^2 = 5476
\][/tex]
5. First Part of the Inequality:
[tex]\[
18.5 \leq \frac{703W}{5476}
\][/tex]
- Multiply both sides by 5476 to isolate [tex]\( W \)[/tex]:
[tex]\[
18.5 \times 5476 \leq 703W
\][/tex]
- Calculate [tex]\( 18.5 \times 5476 \)[/tex]:
[tex]\[
101306 \leq 703W
\][/tex]
- Divide both sides by 703 to solve for [tex]\( W \)[/tex]:
[tex]\[
144.10526315789474 \leq W
\][/tex]
6. Second Part of the Inequality:
[tex]\[
\frac{703W}{5476} \leq 24.9
\][/tex]
- Multiply both sides by 5476 to isolate [tex]\( W \)[/tex]:
[tex]\[
703W \leq 24.9 \times 5476
\][/tex]
- Calculate [tex]\( 24.9 \times 5476 \)[/tex]:
[tex]\[
136256.4 \leq 703W
\][/tex]
- Divide both sides by 703 to solve for [tex]\( W \)[/tex]:
[tex]\[
W \leq 193.9578947368421
\][/tex]
7. Combine the Results:
Combine the lower and upper bounds to specify the weight range:
[tex]\[
144.10526315789474 \leq W \leq 193.9578947368421
\][/tex]
8. Conclusion:
The healthy weight range for a person who is 74 inches tall (6 feet 2 inches) is approximately:
[tex]\[
144.11 \text{ pounds} \leq W \leq 193.96 \text{ pounds}
\][/tex]
This interval represents the healthy weight range based on the given height and BMI boundaries.
