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, 2020

Geometry Difficulty 4.3 AIME Find the answer United States

Problem:

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 $n$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

The $k$ th layer contributes a lateral surface area of $2k\pi$ , so the total lateral surface area is
$2(1+2+\cdots+n)\pi = n(n+1)\pi$
On the other hand, the vertical surface area is $2n^2\pi$ (no need to sum layers, just look at the building from above and from below). Therefore,
$n+1 = \frac{7}{20}(3n+1)$
and $n=13$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.