2021 GMC 10B Problems/Problem 2

Problem

The radius of a circle that has an area of $\frac{\pi}{\sqrt{2}}$ is $r$ . Find $r^{2}$

$\textbf{(A)} ~\frac{1}{2} \qquad\textbf{(B)} ~\frac{\sqrt{2}}{2} \qquad\textbf{(C)} ~\sqrt{2} \qquad\textbf{(D)} ~2 \qquad\textbf{(E)} ~4$

Solution

By the formula for the area of a circle, $\frac{\pi}{\sqrt{2}} = \pi r^2.$

Solving this equation, we get $\begin{align*} \frac{\pi}{\sqrt{2}} &= \pi r^2 \\ \frac{1}{\sqrt{2}} &= r^2 \\ \boxed{\textbf{(B)}~\frac{\sqrt2}{2}} &= r^2. \end{align*}$

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2021 GMC 10B Problems/Problem 2

Problem

Solution