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2021 April MIMC 10 Problems/Problem 6

A worker cuts a piece of wire into two pieces. The two pieces, $A$ and $B$, enclose an equilateral triangle and a square with equal area, respectively. The ratio of the length of $B$ to the length of $A$ can be expressed as $a\sqrt[b]{c}:d$ in the simplest form. Find $a+b+c+d$.

$\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~14 \qquad\textbf{(E)} ~15$

Solution

There are several different ways to solve this problems. For the sake of convenience, we can substitute a side length of the equilateral triangle. Let $2$ be the side length, then the area of the equilateral triangle is $\sqrt{3}$. The side length of the square can be solved by computing $\sqrt[4]{3}$. However, the question is asking for the perimeter of triangle $:$ the perimeter of the square. Therefore, the ratio is $4\cdot{\sqrt[4]{3}:6=2\cdot \sqrt[4]{3}}:3$. $2+4+3+3=\fbox{\textbf{(C)} 12}$.

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