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When we consider the weight of an astronaut, it is important to understand that weight is the force exerted by gravity on a mass. In space, an astronaut's weight changes with their distance from Earth. The formula given in the exercise represents how weight is calculated based on altitude. This formula is based on the principle of physics that the gravitational force diminishes with distance from the Earth's center. It is expressed as: - \( W = 125 \left( \frac{6400}{6400 + x} \right)^2 \) - Here, \( W \) is the weight of the astronaut at altitude \( x \). - The constant 6400 is approximately the Earth's radius in kilometers. - The term \( \left( \frac{6400}{6400 + x} \right)^2 \) represents the fraction of gravitational force acting on the astronaut compared to what they experience at sea level.At sea level (\( x = 0 \)), the equation simplifies and gives the astronaut's full weight of 125 pounds. As \( x \) increases, the fraction becomes smaller, indicating that less gravitational force is acting on the astronaut, thus their weight decreases. Understanding this concept is crucial for astronauts working in different altitudes.

As a space shuttle climbs higher, the altitude affects an astronaut's weight due to decreasing gravitational force. This effect can be computed using the given formula. Let's break down how altitude impacts weight:- An increased altitude \( x \) means moving away from Earth's gravitational pull. - Weight decreases as \( x \) increases because the function value \( \left( \frac{6400}{6400 + x} \right) \) reduces. - For very large \( x \), this fraction approaches zero, reaching a state near weightlessness, not full zero but insignificant compared to sea level.This phenomenon is essential for understanding space missions. It affects how we design space suits and equipment, as astronauts need to account for less weight and potential floating in space. The extremes of altitude stress the importance of calculating weight for safety and mission efficiency.

Inequalities allow us to determine allowable ranges within constraints, a fundamental math concept with practical applications, like in this exercise. The task was to solve for altitudes where the astronaut's weight is less than 5 pounds. Here's how the mathematical journey progresses:1. **Set up the inequality:** - Start with \( 125 \left( \frac{6400}{6400 + x} \right)^2 < 5 \). 2. **Simplify:** - Dividing by 125 and simplifying leads to \( \left( \frac{6400}{6400 + x} \right)^2 < \frac{1}{25} \). 3. **Solving for x:** - By solving \( \frac{6400}{6400 + x} < \frac{1}{5} \) and further operations, isolate \( x \) to get the range \( x > 25600 \).This approach shows how inequalities help us find specific conditions satisfying a requirement. For astronauts, it spells out at which altitude more than negligible weightlessness is achieved, critical for mission planning and astronaut experience in space. Mastering inequalities equips students to solve real-world problems effectively.