Rates of change: inflated hot-air balloon

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SUMMARY

The discussion centers on calculating the rate of change of the diameter of an inflated hot-air balloon modeled as a sphere with a diameter of 30 meters, losing air at a rate of 118 m³/s. The volume of a sphere is expressed as V = (4/3)πr³, and the relationship between volume and diameter is established through the formula V = (1/6)πD³. After 3 seconds, the diameter is decreasing at a specific rate, which can be determined using the chain rule of calculus. The total time for the balloon to completely collapse can also be calculated based on the initial volume and the rate of air loss.

PREREQUISITES
  • Understanding of calculus, specifically the chain rule
  • Knowledge of the volume formula for a sphere, V = (4/3)πr³
  • Familiarity with the relationship between diameter and radius
  • Basic principles of rates of change in physics
NEXT STEPS
  • Study the application of the chain rule in related calculus problems
  • Learn how to derive the volume of a sphere in terms of diameter
  • Explore practical examples of rates of change in real-world scenarios
  • Investigate the physical principles governing the behavior of gases in balloons
USEFUL FOR

Students studying calculus, physics enthusiasts, and anyone interested in understanding rates of change in geometric contexts.

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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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do you know the formula for the volume of a sphere in terms of the radius?
 
4/3∏r^3
 
Then, what is the volume of a sphere in terms of its diameter?
 

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