Chapter 5.5, Problem 3SP

Hot air balloons

Rot all ballooning is a relaxing. peaceful pastime that many people enjoy. Many balloonist gatherings take place around the world, Such as the Albuquerque International Balloon Fiesta. The Albuquerque event is the largest hot air balloon festival in the world, with ovet 500 balloons participating each year.

Figure 5.61 Balloons lift off at the 2001 Albuquerque international Balloon Fiesta. (credit: David Herrera, Flickr)

As the name implies, hot air balloons use hot air to generate lift. (Hot air is less dense than cooler air, so the balloon floats as long as the hot air stays hot.) The heat is generated by a propane burner suspended below the opening of the basket. Once the balloon takes off, the pilot controls the altitude of the balloon, either by using the burner to heat the air and ascend or by using a vent near the top of the balloon to release heated air and descend. The pilot has very little control over where the balloon goes, however—balloons are at the mercy of the winds. The uncertainty over where we will end up is one of the reasons balloonists are attracted to the sport.

In this project we use tnple integrals to learn more about hot air balloons. We model the balloon in two pieces. The top of the balloon is modeled by a half sphere of radius 28 feet. The bottom of the balloon is modeled by a fnistum of a cone (think of an ice cream cone with the pointy end cut off). The radius of the large end of the fnistum is 28 feet and the radius of the small end of the fnistum is 6 feet. A graph of our balloon model and a cross-sectional diagram showing the dimensions are shown in the following figure.

FIgure 5.62 (a)Useahalfsphetetomodeltheroppartoftheballoonandafrnsnunofaconeomodel

the bonom pan of the balloon. (b) A cross section of the balloon shong its dimensions.

3. Find the average temperature of the air in the balloon after the pilot has actIvated the burner br tO seconds.