# Katamari damacy: make a giant ball of crap before time is up!

Calculus Level 5

Katamari Damacy is a popular Japanese video game where a character rolls a ball around a world that is filled with random things. As the ball rolls it picks up objects in its path and engulfs them, resulting in a growth process.

To a first approximation, any object of volume $$V$$ that is picked up is instantly and uniformly distributed to the volume of the growing ball of crap. Also, the ball rolls with constant angular velocity.

Suppose that in a simple world, the character rolls the ball in a straight line that has random crap distributed with a line density of $$\lambda_0$$ kg/m. If the original radius doubles in time $$T_2$$, and it increases eight-fold in time $$T_8$$, what is the value of $$T_8/T_2?$$

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