HMMT 十一月 2020 · THM 赛 · 第 3 题

HMMT November 2020 — THM Round — Problem 3

Harvard has recently built a new house for its students consisting of n levels, where the k th level from the top can be modeled as a 1-meter-tall cylinder with radius k meters. Given that the area of all the lateral surfaces (i.e. the surfaces of the external vertical walls) of the building is 35 percent of the total surface area of the building (including the bottom), compute n .

解析

Harvard has recently built a new house for its students consisting of n levels, where the k th level from the top can be modeled as a 1-meter-tall cylinder with radius k meters. Given that the area of all the lateral surfaces (i.e. the surfaces of the external vertical walls) of the building is 35 percent of the total surface area of the building (including the bottom), compute n . Proposed by: Daniel Zhu Answer: 13 Solution: The k th layer contributes a lateral surface area of 2 kπ , so the total lateral surface area is 2(1 + 2 + · · · + n ) π = n ( n + 1) π. 2 On the other hand, the vertical surface area is 2 n π (No need to sum layers, just look at the building from above and from below). Therefore, 7 n + 1 = (3 n + 1) , 20 and n = 13.