HMMT 二月 2000 · 团队赛 · 第 2 题

HMMT February 2000 — Team Round — Problem 2

The pri e of a gold ring in a ertain univ erse is prop ortional to the square of its purit y and the ub e of its diameter. The purit y is in v ersely prop ortional to the square of the depth of the gold mine and dire tly prop ortional to the square of the pri e, while the diameter is determined so that it is prop ortional to the ub e ro ot of the pri e and also dire tly prop ortional to the depth of the mine. Ho w do es the pri e v ary solely in terms of the depth of the gold mine?

解析

Let =pri e, p =purit y , d =diameter, h =depth of gold mine, k = onstan t. W e are giv en i p 2 4 2 3 2 3 3 5 1 4 3 = k p d , p = k , and d = k h . So, = k k k h = k . Th us, k = h , 1 2 2 3 1 4 4 4 2 3 h h h 1 1 4 4 and = k h . Th us, p v aries as h . 5 799 800 699 700