## Problem of the Day #140: Bessie’s MilkAugust 6, 2011

Posted by Albert in : potd , trackback

Bessie the supercow knows how much milk she can produce on a given day, but wants to test Farmer John’s ability to make containers. She gives FJ a function, $f(d, x)$, which tells FJ that on day $d, (1 <= d < \infty)$, and at linear distance $x, (0 <= x < \infty)$, the container of width $1$ foot has depth $f(d, x)$. FJ, being the clever farmer that he is, will not make it, instead constructing a simpler container with the same volume.

Unfortunately, Bessie’s function gets distorted in transmission, and several constants are just boxes: $$f(d, x) = \Box x^{10} d^{20} \Box^{-d^2 x^7}$$ Having no idea what to do, on the first day ($d = 1$) he goes out with a large container and milks Bessie, getting 1 gallon of milk. FJ then realizes he can figure out how many gallons are needed on any given day.

Let the volume on day $100 = a^{\frac{b}{c}}$, where $a$, $b$, $c$ are positive integers, $GCD(b, c) = 1$, $a$ is as low as possible, find the concatenation of $a$, $b$, and $c$.