2. The design of the Trieste was based on the design of a hot air balloen built by record breaking height. a. Auguste's ascent took 7 hours and went up 51,683 feet. Write a relationship \( y=k x \) to represent his ascent from his starting location. b. Auguste's descent took 3 hours and went down 52,940 feet. Write another relationship to represent his descent. C.Did Auguste Piccard end up at a greater or lesser altitude than his starting point? How much higher or lower? Are you ready for more? During which part of either trip was a Piccard changing vertical position the fastest? Explain your reasoning. - Jacques's descent - Jacques's ascent - Auguste's ascent - Auguste's descent Lesson 12 Summary We saw earller that we can represent speed with direction using signed numbers. Speed with direction is called velocity. Positive velocities always represent movement in the opposite direction from negative velocities. We can do this with vertical movement: moving up can be represented with positive numbers, and moving down with negative numbers. The magnitude tells you how fast, and the sign tells you which direction. (We could actually do it the other way around if we wanted to, but usually we make up positive and down negative.)

Let's break down the problem step by step. ### Part a: Ascent Relationship 1. **Known Conditions**: - Ascent time: 7 hours - Ascent height: 51,683 feet 2. **Relationship**: We can express the ascent as a linear relationship \( y = kx \), where: - \( y \) is the height in feet, - \( x \) is the time in hours, - \( k \) is the rate of ascent in feet per hour. 3. **Calculate \( k \)**: \[ k = \frac{\text{height}}{\text{time}} = \frac{51683 \text{ feet}}{7 \text{ hours}} \approx 7397.57 \text{ feet/hour} \] 4. **Final Relationship**: The relationship for the ascent is: \[ y = 7397.57x \] ### Part b: Descent Relationship 1. **Known Conditions**: - Descent time: 3 hours - Descent height: 52,940 feet 2. **Relationship**: We can express the descent as a linear relationship \( y = kx \), where: - \( y \) is the height in feet (negative for descent), - \( x \) is the time in hours, - \( k \) is the rate of descent in feet per hour. 3. **Calculate \( k \)**: \[ k = \frac{-52940 \text{ feet}}{3 \text{ hours}} \approx -17646.67 \text{ feet/hour} \] 4. **Final Relationship**: The relationship for the descent is: \[ y = -17646.67x \] ### Part c: Final Altitude Comparison 1. **Final Altitude Calculation**: - Starting altitude: 0 feet - Final altitude after ascent: \( 51683 \) feet - Final altitude after descent: \( 51683 - 52940 = -1257 \) feet 2. **Comparison**: Auguste Piccard ended up at a lesser altitude than his starting point. He is 1,257 feet lower than his starting altitude. ### Part d: Fastest Vertical Position Change To determine during which part of the trip Piccard was changing vertical position the fastest, we compare the rates of ascent and descent: - **Auguste's ascent**: \( 7397.57 \) feet/hour - **Auguste's descent**: \( 17646.67 \) feet/hour (in magnitude) Since the descent rate is greater in magnitude than the ascent rate, Auguste was changing vertical position the fastest during his descent. ### Summary of Answers: - **Ascent Relationship**: \( y = 7397.57x \) - **Descent Relationship**: \( y = -17646.67x \) - **Final Altitude**: 1,257 feet lower than starting point. - **Fastest Vertical Change**: Auguste's descent.