Lenny Conundrum Round 381

Faerieland is a wonderful world that rests on magical clouds that float over Neopia. (Well, at least, they used to float over Neopia...)

These clouds are made of a special magical lightweight polymer material that supports the weight of the world on top of it. The clouds weigh 63.4 kilograms per cubic metre. And the cloud is a massive hemisphere, 3450 metres in diameter. Also, assume all of the buildings, faeries, and other objects on top of the cloud weigh 750,000 kg.

If one faerie's magic can cause 123,456 kg of weight to float in the sky, how many faeries are required to keep all of Faerieland floating? Disregard any buoyancy effect from the surrounding air, and be sure to round up to the nearest whole number.

Possible Methods:

Volume of the cloud = 2/3*pi*r³

Total number of faeries required = (Volume of the cloud * Cloud density + Weight on top of the cloud) / Weight that one faerie's magic can cause to float

= (2/3*pi*(3450/2)³ * 63.4 + 750000) / 123456

= 5520818.2

= 5520819 (Rounded up to the nearest whole number)

Answer: 5520819

 Lenny Conundrum Round 372

Moltara is a very hot place, and the demand for cold water has been increasing ever since more people started to visit. Therefore, the people of Moltara decided to build a new water tank.

This water tank is a perfect sphere with an inner radius of 12 metres. However, they used inferior materials to build the water tank, and the tank can only withstand a water pressure of 185 kiloPascals. Assume the density of water is exactly 1000 kg/m³, gravity is 9.8 metres/s², and disregard atmospheric pressure.

How many cubic metres of water can safely be stored in the tank? Round DOWN to the nearest whole number (since rounding up could be catastrophic in this instance!), and please submit only a number with no other information.

Possible Methods:

Max Pressure = Density * Max Height * Gravity

Max Height = 185 * 10³ / 1000 / 9.8 = 185 / 9.8 m

Cap Height, h = Diameter - Max Height = 12 * 2 - 185 / 9.8 = 24 - 185 / 9.8 m

Volume of Water that can be stored = 4/3 * pi * r³ - 1/3 * pi * h² * (3r - h)

= 4/3 * pi * 12³ - 1/3 * pi * (24 - 185 / 9.8)² * (3 * 12 - 24 + 185 / 9.8)

= 6389.78 m^3

Answer: 6389