Tuesday, May 17, 2005

Math Problem VII

1) A cubic meter of helium gas can lift a load of one kilogram. Assume that the mass of the balloon (and string or ribbon) is negligible, and calculate the diameter of the spherical balloon required to lift:

A) a 21-gram mouse
B) a nine pound, two ounce newborn baby in a fifteen-pound car seat
C) a 230-pound giant panda (Ailuropoda melanoleuca)
D) a Humvee (5200 pounds)

Extra Credit 1: an average-sized rubber party balloon has approximately the same volume as a sphere eleven inches in diameter. Determine for each of the above how many such helium-filled balloons would be required to lift the object in question.

Extra Credit 2: a king-size waterbed mattress contains about 1.325 cubic meters of water when full. Determine for each of the above how many such helium-filled mattresses would be required to lift the object in question.

2) Annual world production of helium (mostly from natural gas) is 100 million cubic meters. How heavy of an object could this lift if it were all gathered together in a single spherical balloon?

conversion factors:
1 cubic meter = 1000 liters
1 pound = 16 ounces
1 pound = 2.2046 kilograms
1 kilogram = 1000 grams
1 ounce = 28.35 grams
1 foot = 0.3048 meters
1 inch = 2.54 centimeters
1 foot = 12 inches
1 meter = 100 centimeters

formulas:
volume of a sphere = (4π/3)r3, where r is the radius
diameter of a sphere = 2r, where r is the radius

constant:
π = 3.14159

Answers in comments.

2 comments:

Jessi Guilford said...

1) A) a 21-gram mouse:

(21 grams load)(1 m^3 He/1 kg load)(1 kg / 1000 g) = 0.021 m^3 He

(0.021 m^3 He) = (4π/3)r^3

r= [(0.021)(3)/4π]^(1/3) = 0.171 m

(0.171 m)(100 cm / m)(1 inch / 2.54 cm) = 6.73 inches = radius

(6.73 inches)(2) = diameter = 13.46 inches

It would therefore take a spherical helium-filled balloon about 13.5 inches in diameter to lift a 21-gram mouse. This is approximately the volume of the standard helium-filled rubber party balloon.

B) a nine pound, two ounce newborn baby in a fifteen-pound car seat:

9 lb. + 2/16 lb. + 15 lb = 24.125 lb.

(24.125 lb. load)(1 m^3 He / 1 kg load)(2.2046 kg / lb.) = 53.186 m^3 He

by a similar process to part A), we get a radius of 2.333 meters, and a diameter of 4.666 meters, which converts to a diameter of
(4.666 meters)(1 foot / 0.3048 meters) = 15.31 feet.

It would therefore take a spherical helium-filled balloon about 15.31 feet in diameter to lift a newborn baby in a carseat. This is equivalent to about 4,660 balloons, or about 40 king-size waterbed mattresses.

C) a 230-pound giant panda

as in B): (230)(2.2046) = 507 m^3 He

which converts to a diameter of 389.5 inches, or 32.5 feet. This is the equivalent of 44,400 party balloons or 383 waterbed mattresses.

D) a Humvee

as in B): (5200)(2.2046) =11463.92 m^3 He

which converts to a diameter of 1101.4 inches, or 91.8 feet. This is the equivalent of 1,004,000 balloons or 8,650 waterbed mattresses.

2)

(100,000,000 m^3 He)(1 kg / 1 m^3 He) = 100,000,000 kg load. This is the weight of about 1.4 million people. 1.4 million people is:

the estimated population of diabetics in the UK;
the weekly increase in world population;
the approximate population of Phoenix, Arizona,
the approximate population of the U.S. state of Idaho;
the approximate population of the U.S. city of Nashville, Tennessee;
the approximate population of the U.S. state of Maine;
the approximate population of the U.S. state of New Hampshire;
the approximate population of the Canadian province of Manitoba;
the approximate population of Copenhagen, Denmark;
the approximate population of Dusseldorf, Germany;
the approximate population of Austin, Texas;
the approximate population of Salt Lake City, Utah;
the population newly diagnosed with tuberculosis in China each year; or
the Jewish population in the New York City metropolitan area.

Rory said...

no! my sister was a small child, and one year when the parents actually hired a helium cylinder to go with the cake and pinata, we tied at least fifty balloons to her, and she wouldn't drift off in the lazy thermals to frnace. oh no. she's still around. and incidentally, i live in guildford, which i'm sure you didnt know about, and i'm not some kind of louse. your site is brilliant.