# 1950 AHSME Problems/Problem 33

## Problem

The number of circular pipes with an inside diameter of $1$ inch which will carry the same amount of water as a pipe with an inside diameter of $6$ inches is: $\textbf{(A)}\ 6\pi \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 36\pi$

## Solution

It must be assumed that the pipes have an equal height.

A circular pipe with diameter 1 inch and height h has a volume of $\pi \left(\frac{1}{2}\right)^2h=\frac{\pi h}{4}$. A pipe with diameter 6 inches and height h has volume $\pi \left(\frac{6}{2}\right)^2h=9\pi h$. To find how many 1-pipes fit in a 6-pipe, we divide: $\frac{9\pi h}{\frac{\pi h}{4}}=\frac{9*4\pi h}{\pi h}=\frac{36\pi h}{\pi h}=36 \textbf{(D)}$

If the ratio of similar length of similar shapes is x, then the ratio between area is $x^2$. Therefore, since the ratio between diameters is $1/6$, the ratio between area is $1/36$, so $36= \textbf{(D)}$ pipes of diameter $1$ are required.

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