M19-21

Official Solution:

If the area of a square equals the area of a circle, which of the following is closest to the ratio of the diagonal of the square to the diameter of the circle?

A. 0.95
B. 1.26
C. 1.40
D. 1.57
E. 2.51

If \(x\) is the side of the square, then the area of the square is \(x^2\) and the diagonal of the square is \(x\sqrt{2}\). If \(d\) is the diameter of the circle then the area of the circle is \(\pi(\frac{d}{2})^2\). Because \(x^2 = \pi(\frac{d}{2})^2\), \(d = \frac{2x}{\sqrt{\pi}}\). The required ratio \(=\frac{x\sqrt{2}}{\frac{2x}{\sqrt{\pi}}} = \frac{\sqrt{\pi}}{\sqrt{2}} = \sqrt{\frac{\pi}{2}}\) or approximately \(\sqrt{1.57}\). This is slightly smaller than \(\sqrt{1.69} = 1.3\). The best answer is therefore B.

Answer: B