Glass Half Full

**I am trying to find all the solutions and put them together at a single place, So feel free to add comments and if you have a solution please do post. Cheers … :)**

**Question :** You are in an empty room and you have a transparent glass of water. The glass is a right cylinder, and it looks like it’s half full, but you’re not sure. How can you accurately figure out whether the glass is half full, more than half full, or less than half full? You have no rulers or writing utensils.

Hint 1: To help you get started if you’re stuck, here’s a solution that’s not good enough. Holding the cup upright, use the palm of your left hand to cover the cup’s opening. Now make a pinching gesture with the index finger and thumb of your right hand. Put the thumb at the base of the cup, and the index finger adjacent to the water level, thereby gauging the height of the water surface from the base of the cup. Now freeze the distance between those two fingers. Flip the cup upside down with your left hand; no water falls out since you’ve sealed the opening with your left palm. Now put your frozen right hand against the cup, and see if the inverted water level is next to your index finger. If so, the cup is exactly half full. This seems like a good solution, but it’s actually slightly accurate, because the palm of your hand is not a perfectly flat surface. Also, you’ll most likely lose some water when you flip the cup upside down. We want a really accurate method.

Hint 2: Utilize the geometry of the cup. That’s really important.

Hint 3: Don’t drink the water. ^_^

**My Solution : **Tilt the glass till the water comes upto the brim. If the glass is exactly half filled, then the 2 ends of the water will be the brim and the upper bottom of the tilted glass. If the water is less than half, the water level will be lower than the upper bottom of the tilted glass and vice versa. This is possible because, the glass is a right circular cylinder, which means that the length of the radius is equal to the height of the cylinder. Another explanations is through the volumes. If the glass is half full, then the volume of the water must be ∏ rˆ3/2 . And if the tilted glass has the water level touching the brim and the upper bottom of the glass then their volume will also be V/2 .

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