Rates of change: inflated hot-air balloon

In summary, a hot air balloon with a diameter of 30m loses hot air at a rate of 118m^3/s when the hatch is opened. After 5 seconds, the balloon begins to collapse and the rate at which its diameter decreases is unknown. The question is asking for the formula to relate the rate of change of volume to the rate of change of diameter. The formula for the volume of a sphere in terms of the radius is 4/3∏r^3, and in terms of the diameter it is (4/3∏(d/2)^3).
  • #1
jackscholar
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Homework Statement


An inflated hot air balloon can be considered as a perfect sphere woth diameter of 30m. When the hatch is opened, the hot air is lost at a rate of 118m^3/s. For the first 5 seconds, the balloon maintains its spherical shape, but then it begins to collapse. At what rate is the diameter decreasing after 3 seconds? How long does the balloon take to completely collapse?

I know this is a chain rule question where dV/dt=118m^3/s and I'm not sure how to relate that to the diameter? I know that, because dV/dt is part of the equation the equation is either dV/dt=dV/dD*dD/dt where D is the diameter or the equation dD/dt=dD/dV*dV/dt or am I going about this the completely wrong way and have to get everything in terms of the radius?
 
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  • #2
do you know the formula for the volume of a sphere in terms of the radius?
 
  • #3
4/3∏r^3
 
  • #4
Then, what is the volume of a sphere in terms of its diameter?
 

Related to Rates of change: inflated hot-air balloon

1. What is the concept of rates of change in relation to inflated hot-air balloons?

The concept of rates of change refers to the measurement of how quickly or slowly a variable, such as the size or altitude of an inflated hot-air balloon, is changing over time. It can be represented by a slope on a graph or calculated using mathematical equations.

2. How does the rate of change affect the flight of an inflated hot-air balloon?

The rate of change can significantly impact the flight of an inflated hot-air balloon. For example, if the rate of change of the balloon's altitude is too high, it could lead to rapid ascents or descents, which could be dangerous for the passengers on board. Additionally, a constant rate of change is necessary for maintaining a steady and controlled flight.

3. What factors can influence the rate of change of an inflated hot-air balloon?

There are several factors that can affect the rate of change of an inflated hot-air balloon, including the temperature of the air inside and outside the balloon, the weight of the passengers and cargo, and the strength and direction of the wind. These factors can cause the balloon to rise or fall at different rates, affecting its flight path.

4. How can the rate of change of an inflated hot-air balloon be controlled?

The rate of change of an inflated hot-air balloon can be controlled through various methods, such as adjusting the temperature of the air inside the balloon, changing the amount of weight on board, and using directional tools like a rudder or burner. Pilots must constantly monitor and make appropriate adjustments to maintain a safe and stable flight.

5. How does the rate of change of an inflated hot-air balloon impact its landing?

The rate of change of an inflated hot-air balloon is crucial during its landing. A controlled rate of descent is necessary to ensure a safe and smooth landing, while a sudden or rapid change in altitude could result in a crash. Pilots must carefully manage the rate of change during landing to ensure the safety of all passengers on board.

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